didericis/math-research
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

Add small-n ELG enumeration table to even_level_graph_generators

Browse Source 
Records, for 4<=n<=11, triangulation iso classes, how many admit an ELG
source, ELG iso classes, and the automorphism-free flag-rooted count
sum_G 4E/|Aut(G)| * s(G). Computed by experiments/count_elgs.py.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-22 12:24:07 -04:00
parent 8fde9494d8
commit 4693f63208
7 changed files with 235 additions and 268 deletions
Download Patch File Download Diff File Expand all files Collapse all files
papers/even_level_graph_generators/paper.aux
+13 -10
View File
@@ -34,23 +34,26 @@
\newlabel{sec:even-level-graphs}{{4}{4}{Even Level Graphs}{section.4}{}} \newlabel{sec:even-level-graphs}{{4}{4}{Even Level Graphs}{section.4}{}}
\newlabel{def:even-level-graph}{{4.1}{4}{Even Level Graph}{theorem.4.1}{}} \newlabel{def:even-level-graph}{{4.1}{4}{Even Level Graph}{theorem.4.1}{}}
\newlabel{thm:even-level-4colorable}{{4.2}{4}{}{theorem.4.2}{}} \newlabel{thm:even-level-4colorable}{{4.2}{4}{}{theorem.4.2}{}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Enumeration for small $n$}}{5}{section*.1}\protected@file@percent }
\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Even Level Graph counts for $4 \leq n \leq 11$. The \emph {triangulations} column is the number of plane-triangulation iso classes (OEIS A000109). \emph {ELG iso classes} counts pairs $(G, S)$ up to isomorphism; \emph {flag-rooted ELGs} is the automorphism-free count $\DOTSB \sum@ \slimits@ _G \genfrac {}{}{}1{4E}{|\mathrm {Aut}(G)|}\,s(G)$.}}{5}{table.1}\protected@file@percent }
\newlabel{tab:elg-counts}{{1}{5}{Even Level Graph counts for $4 \leq n \leq 11$. The \emph {triangulations} column is the number of plane-triangulation iso classes (OEIS A000109). \emph {ELG iso classes} counts pairs $(G, S)$ up to isomorphism; \emph {flag-rooted ELGs} is the automorphism-free count $\sum _G \tfrac {4E}{|\mathrm {Aut}(G)|}\,s(G)$}{table.1}{}}
\newlabel{def:derived-level-graph}{{4.3}{5}{Derived level graph}{theorem.4.3}{}} \newlabel{def:derived-level-graph}{{4.3}{5}{Derived level graph}{theorem.4.3}{}}
\newlabel{def:bridge-switch}{{4.4}{5}{Bridge switch}{theorem.4.4}{}}
\newlabel{def:bridge-derived-level-graph}{{4.5}{5}{Bridge-derived level graph}{theorem.4.5}{}}
\newlabel{def:intertwining-tree}{{4.6}{5}{Intertwining tree}{theorem.4.6}{}}
\newlabel{thm:intertwining-iff-hamiltonian-dual}{{4.7}{5}{}{theorem.4.7}{}}
\citation{holton-mckay} \citation{holton-mckay}
\newlabel{def:bridge-switch}{{4.4}{6}{Bridge switch}{theorem.4.4}{}}
\newlabel{def:bridge-derived-level-graph}{{4.5}{6}{Bridge-derived level graph}{theorem.4.5}{}}
\newlabel{def:intertwining-tree}{{4.6}{6}{Intertwining tree}{theorem.4.6}{}}
\newlabel{thm:intertwining-iff-hamiltonian-dual}{{4.7}{6}{}{theorem.4.7}{}}
\newlabel{conj:every-triangulation-derived}{{4.8}{6}{}{theorem.4.8}{}} \newlabel{conj:every-triangulation-derived}{{4.8}{6}{}{theorem.4.8}{}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The boundary case $n = 21$}}{6}{section*.1}\protected@file@percent }
\bibcite{holton-mckay}{1} \bibcite{holton-mckay}{1}
\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces The six Holton--McKay duals at $n = 21$, the first triangulations that are not intertwining trees. Each is a bridge-derived level graph: duals $1$ and $2$ are Even Level Graphs outright (zero switches), and the remaining four reach an Even Level Graph in $1$--$4$ bridge switches. All witnesses are step-verified.}}{7}{table.1}\protected@file@percent }
\newlabel{tab:n21}{{1}{7}{The six Holton--McKay duals at $n = 21$, the first triangulations that are not intertwining trees. Each is a bridge-derived level graph: duals $1$ and $2$ are Even Level Graphs outright (zero switches), and the remaining four reach an Even Level Graph in $1$--$4$ bridge switches. All witnesses are step-verified}{table.1}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The six Holton--McKay duals, drawn as crossing-free planar graphs and coloured by parity (blue even, orange odd, with respect to the fixed level-parity labelling). The solid green edges are the bridge edges introduced by the bridge switches from each dual's witness Even Level Graph. Each green edge is a bridge of its parity subgraph, so no new cycle -- and in particular no odd cycle -- is created; duals $1$ and $2$ coincide with their Even Level Graphs and have no added edge.}}{7}{figure.5}\protected@file@percent }
\newlabel{fig:n21-duals}{{5}{7}{The six Holton--McKay duals, drawn as crossing-free planar graphs and coloured by parity (blue even, orange odd, with respect to the fixed level-parity labelling). The solid green edges are the bridge edges introduced by the bridge switches from each dual's witness Even Level Graph. Each green edge is a bridge of its parity subgraph, so no new cycle -- and in particular no odd cycle -- is created; duals $1$ and $2$ coincide with their Even Level Graphs and have no added edge}{figure.5}{}}
\newlabel{tocindent-1}{0pt} \newlabel{tocindent-1}{0pt}
\newlabel{tocindent0}{14.69437pt} \newlabel{tocindent0}{14.69437pt}
\newlabel{tocindent1}{17.77782pt} \newlabel{tocindent1}{17.77782pt}
\newlabel{tocindent2}{0pt} \newlabel{tocindent2}{0pt}
\newlabel{tocindent3}{0pt} \newlabel{tocindent3}{0pt}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{8}{section*.2}\protected@file@percent } \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The boundary case $n = 21$}}{7}{section*.2}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{7}{section*.3}\protected@file@percent }
\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces The six Holton--McKay duals at $n = 21$, the first triangulations that are not intertwining trees. Each is a bridge-derived level graph: duals $1$ and $2$ are Even Level Graphs outright (zero switches), and the remaining four reach an Even Level Graph in $1$--$4$ bridge switches. All witnesses are step-verified.}}{8}{table.2}\protected@file@percent }
\newlabel{tab:n21}{{2}{8}{The six Holton--McKay duals at $n = 21$, the first triangulations that are not intertwining trees. Each is a bridge-derived level graph: duals $1$ and $2$ are Even Level Graphs outright (zero switches), and the remaining four reach an Even Level Graph in $1$--$4$ bridge switches. All witnesses are step-verified}{table.2}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The six Holton--McKay duals, drawn as crossing-free planar graphs and coloured by parity (blue even, orange odd, with respect to the fixed level-parity labelling). The solid green edges are the bridge edges introduced by the bridge switches from each dual's witness Even Level Graph. Each green edge is a bridge of its parity subgraph, so no new cycle -- and in particular no odd cycle -- is created; duals $1$ and $2$ coincide with their Even Level Graphs and have no added edge.}}{8}{figure.5}\protected@file@percent }
\newlabel{fig:n21-duals}{{5}{8}{The six Holton--McKay duals, drawn as crossing-free planar graphs and coloured by parity (blue even, orange odd, with respect to the fixed level-parity labelling). The solid green edges are the bridge edges introduced by the bridge switches from each dual's witness Even Level Graph. Each green edge is a bridge of its parity subgraph, so no new cycle -- and in particular no odd cycle -- is created; duals $1$ and $2$ coincide with their Even Level Graphs and have no added edge}{figure.5}{}}
\gdef \@abspage@last{8} \gdef \@abspage@last{8}
papers/even_level_graph_generators/paper.fdb_latexmk
+5 -30
View File
@@ -1,25 +1,6 @@
# Fdb version 3 # Fdb version 3
["pdflatex"] 1779464918 "/Users/didericis/Code/math-research/papers/even_level_graph_generators/paper.tex" "paper.pdf" "paper" 1779464920 ["pdflatex"] 1779466997 "/Users/didericis/Code/math-research/papers/even_level_graph_generators/paper.tex" "paper.pdf" "paper" 1779466999
"/Users/didericis/Code/math-research/papers/even_level_graph_generators/paper.tex" 1779464918 21409 f57d3b5341eb97db9586774d14c6260d "" "/Users/didericis/Code/math-research/papers/even_level_graph_generators/paper.tex" 1779466995 23442 551d7ef123af98c3c5f0ca2b7125c5d5 ""
"/usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1246382020 1004 54797486969f23fa377b128694d548df ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1246382020 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm" 1246382020 916 f87d7c45f9c908e672703b83b72241a3 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam5.tfm" 1246382020 924 9904cf1d39e9767e7a3622f2a125a565 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam7.tfm" 1246382020 928 2dc8d444221b7a635bb58038579b861a ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm" 1246382020 908 2921f8a10601f252058503cc6570e581 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm5.tfm" 1246382020 940 75ac932a52f80982a9f8ea75d03a34cf ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm" 1246382020 940 228d6584342e91276bf566bcf9716b83 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmbx10.tfm" 1136768653 1328 c834bbb027764024c09d3d2bf908b5f0 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmcsc10.tfm" 1136768653 1300 63a6111ee6274895728663cf4b4e7e81 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi6.tfm" 1136768653 1512 f21f83efb36853c0b70002322c1ab3ad ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi8.tfm" 1136768653 1520 eccf95517727cb11801f4f1aee3a21b4 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr6.tfm" 1136768653 1300 b62933e007d01cfd073f79b963c01526 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr8.tfm" 1136768653 1292 21c1c5bfeaebccffdb478fd231a0997d ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy6.tfm" 1136768653 1116 933a60c408fc0a863a92debe84b2d294 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy8.tfm" 1136768653 1120 8b7d695260f3cff42e636090a8002094 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti10.tfm" 1136768653 1480 aa8e34af0eb6a2941b776984cf1dfdc4 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti8.tfm" 1136768653 1504 1747189e0441d1c18f3ea56fafc1c480 ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb" 1248133631 34811 78b52f49e893bcba91bd7581cdc144c0 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb" 1248133631 34811 78b52f49e893bcba91bd7581cdc144c0 ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb" 1248133631 32001 6aeea3afe875097b1eb0da29acd61e28 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb" 1248133631 32001 6aeea3afe875097b1eb0da29acd61e28 ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb" 1248133631 30251 6afa5cb1d0204815a708a080681d4674 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb" 1248133631 30251 6afa5cb1d0204815a708a080681d4674 ""
@@ -39,7 +20,6 @@
"/usr/local/texlive/2022/texmf-dist/tex/generic/bigintcalc/bigintcalc.sty" 1576625341 40635 c40361e206be584d448876bba8a64a3b "" "/usr/local/texlive/2022/texmf-dist/tex/generic/bigintcalc/bigintcalc.sty" 1576625341 40635 c40361e206be584d448876bba8a64a3b ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/bitset/bitset.sty" 1576016050 33961 6b5c75130e435b2bfdb9f480a09a39f9 "" "/usr/local/texlive/2022/texmf-dist/tex/generic/bitset/bitset.sty" 1576016050 33961 6b5c75130e435b2bfdb9f480a09a39f9 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/etexcmds/etexcmds.sty" 1576625273 7734 b98cbb34c81f667027c1e3ebdbfce34b "" "/usr/local/texlive/2022/texmf-dist/tex/generic/etexcmds/etexcmds.sty" 1576625273 7734 b98cbb34c81f667027c1e3ebdbfce34b ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/gettitlestring/gettitlestring.sty" 1576625223 8371 9d55b8bd010bc717624922fb3477d92e ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/iftex/iftex.sty" 1644112042 7237 bdd120a32c8fdb4b433cf9ca2e7cd98a "" "/usr/local/texlive/2022/texmf-dist/tex/generic/iftex/iftex.sty" 1644112042 7237 bdd120a32c8fdb4b433cf9ca2e7cd98a ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/infwarerr/infwarerr.sty" 1575499628 8356 7bbb2c2373aa810be568c29e333da8ed "" "/usr/local/texlive/2022/texmf-dist/tex/generic/infwarerr/infwarerr.sty" 1575499628 8356 7bbb2c2373aa810be568c29e333da8ed ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/intcalc/intcalc.sty" 1576625065 31769 002a487f55041f8e805cfbf6385ffd97 "" "/usr/local/texlive/2022/texmf-dist/tex/generic/intcalc/intcalc.sty" 1576625065 31769 002a487f55041f8e805cfbf6385ffd97 ""
@@ -52,7 +32,6 @@
"/usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls" 1591045760 61881 a7369c346c2922a758ae6283cc1ed014 "" "/usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls" 1591045760 61881 a7369c346c2922a758ae6283cc1ed014 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty" 1359763108 5949 3f3fd50a8cc94c3d4cbf4fc66cd3df1c "" "/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty" 1359763108 5949 3f3fd50a8cc94c3d4cbf4fc66cd3df1c ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd" 1359763108 961 6518c6525a34feb5e8250ffa91731cff "" "/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd" 1359763108 961 6518c6525a34feb5e8250ffa91731cff ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd" 1359763108 961 d02606146ba5601b5645f987c92e6193 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty" 1622667781 2222 da905dc1db75412efd2d8f67739f0596 "" "/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty" 1622667781 2222 da905dc1db75412efd2d8f67739f0596 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty" 1622667781 4173 bc0410bcccdff806d6132d3c1ef35481 "" "/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty" 1622667781 4173 bc0410bcccdff806d6132d3c1ef35481 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty" 1636758526 87648 07fbb6e9169e00cb2a2f40b31b2dbf3c "" "/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty" 1636758526 87648 07fbb6e9169e00cb2a2f40b31b2dbf3c ""
@@ -73,18 +52,15 @@
"/usr/local/texlive/2022/texmf-dist/tex/latex/hycolor/hycolor.sty" 1580250785 17914 4c28a13fc3d975e6e81c9bea1d697276 "" "/usr/local/texlive/2022/texmf-dist/tex/latex/hycolor/hycolor.sty" 1580250785 17914 4c28a13fc3d975e6e81c9bea1d697276 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/hpdftex.def" 1645564115 49029 7c9e5115b2217efbeb7828ac0d1bf1a0 "" "/usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/hpdftex.def" 1645564115 49029 7c9e5115b2217efbeb7828ac0d1bf1a0 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/hyperref.sty" 1645564115 220999 6145ea83914c186e178d1d31c50b37df "" "/usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/hyperref.sty" 1645564115 220999 6145ea83914c186e178d1d31c50b37df ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/nameref.sty" 1612734870 13242 133e617c5eebffdd05e421624022b267 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/pd1enc.def" 1645564115 14132 e8e7e61e51ade521a7238fac8362786c "" "/usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/pd1enc.def" 1645564115 14132 e8e7e61e51ade521a7238fac8362786c ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/puenc.def" 1645564115 117004 ed1c2cc82bb9836e9d59549dd8c33098 "" "/usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/puenc.def" 1645564115 117004 ed1c2cc82bb9836e9d59549dd8c33098 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/kvoptions/kvoptions.sty" 1602274869 22521 d2fceb764a442a2001d257ef11db7618 "" "/usr/local/texlive/2022/texmf-dist/tex/latex/kvoptions/kvoptions.sty" 1602274869 22521 d2fceb764a442a2001d257ef11db7618 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def" 1644269979 29921 d0acc05a38bd4aa3af2017f0b7c137ce "" "/usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def" 1644269979 29921 d0acc05a38bd4aa3af2017f0b7c137ce ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg" 1279039959 678 4792914a8f45be57bb98413425e4c7af "" "/usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg" 1279039959 678 4792914a8f45be57bb98413425e4c7af ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/letltxmacro/letltxmacro.sty" 1575499565 5766 13a9e8766c47f30327caf893ece86ac8 "" "/usr/local/texlive/2022/texmf-dist/tex/latex/letltxmacro/letltxmacro.sty" 1575499565 5766 13a9e8766c47f30327caf893ece86ac8 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/refcount/refcount.sty" 1576624809 9878 9e94e8fa600d95f9c7731bb21dfb67a4 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/rerunfilecheck/rerunfilecheck.sty" 1575674187 9715 b051d5b493d9fe5f4bc251462d039e5f "" "/usr/local/texlive/2022/texmf-dist/tex/latex/rerunfilecheck/rerunfilecheck.sty" 1575674187 9715 b051d5b493d9fe5f4bc251462d039e5f ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/url/url.sty" 1388531844 12796 8edb7d69a20b857904dd0ea757c14ec9 "" "/usr/local/texlive/2022/texmf-dist/tex/latex/url/url.sty" 1388531844 12796 8edb7d69a20b857904dd0ea757c14ec9 ""
"/usr/local/texlive/2022/texmf-dist/web2c/texmf.cnf" 1646502317 40171 cdab547de63d26590bebb3baff566530 "" "/usr/local/texlive/2022/texmf-dist/web2c/texmf.cnf" 1646502317 40171 cdab547de63d26590bebb3baff566530 ""
"/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map" 1647878959 4410336 7d30a02e9fa9a16d7d1f8d037ba69641 ""
"/usr/local/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt" 1665017617 2826443 7e98410c533054b636c6470db83a27bc "" "/usr/local/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt" 1665017617 2826443 7e98410c533054b636c6470db83a27bc ""
"/usr/local/texlive/2022/texmf.cnf" 1647878952 577 209b46be99c9075fd74d4c0369380e8c "" "/usr/local/texlive/2022/texmf.cnf" 1647878952 577 209b46be99c9075fd74d4c0369380e8c ""
"fig_edge_switch.png" 1779389598 123855 9cba07bb97424e0c4d80a584255b6e57 "" "fig_edge_switch.png" 1779389598 123855 9cba07bb97424e0c4d80a584255b6e57 ""
@@ -92,11 +68,10 @@
"fig_levels.png" 1779389598 88029 5564f46c0a183f3777727b651e7dc461 "" "fig_levels.png" 1779389598 88029 5564f46c0a183f3777727b651e7dc461 ""
"fig_parity_subgraph.png" 1779389598 191771 f069aa94c8f49b3c7fd9c71426feff2d "" "fig_parity_subgraph.png" 1779389598 191771 f069aa94c8f49b3c7fd9c71426feff2d ""
"figures/n21_duals.png" 1779463364 667947 fd52170c20399b0c2dff901831fad5d5 "" "figures/n21_duals.png" 1779463364 667947 fd52170c20399b0c2dff901831fad5d5 ""
"paper.aux" 1779464919 7535 db1f66072ebbe4d2fdf51a9fe4ad8e97 "pdflatex" "paper.aux" 1779466999 8486 a3527cbc146de3799e0a68e288b7304f ""
"paper.out" 1779464919 911 06539e3751da0b503943ca6640b71438 "pdflatex" "paper.out" 1779466999 1088 cf07a31709ba02be3ba2bc89322768d0 "pdflatex"
"paper.tex" 1779464918 21409 f57d3b5341eb97db9586774d14c6260d "" "paper.tex" 1779466995 23442 551d7ef123af98c3c5f0ca2b7125c5d5 ""
(generated) (generated)
"paper.aux"
"paper.log" "paper.log"
"paper.out" "paper.out"
"paper.pdf" "paper.pdf"
papers/even_level_graph_generators/paper.fls
+1 -147
View File
@@ -2,7 +2,7 @@ PWD /Users/didericis/Code/math-research/papers/even_level_graph_generators
INPUT /usr/local/texlive/2022/texmf.cnf INPUT /usr/local/texlive/2022/texmf.cnf
INPUT /usr/local/texlive/2022/texmf-dist/web2c/texmf.cnf INPUT /usr/local/texlive/2022/texmf-dist/web2c/texmf.cnf
INPUT /usr/local/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt INPUT /usr/local/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt
INPUT /Users/didericis/Code/math-research/papers/even_level_graph_generators/paper.tex INPUT paper.tex
OUTPUT paper.log OUTPUT paper.log
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
@@ -442,149 +442,3 @@ INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.de
INPUT ./paper.aux INPUT ./paper.aux
INPUT paper.aux INPUT paper.aux
INPUT paper.aux INPUT paper.aux
OUTPUT paper.aux
INPUT /usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr8.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr6.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi8.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi6.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy8.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy6.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam7.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam5.tfm
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm5.tfm
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/nameref.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/nameref.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/nameref.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/nameref.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/nameref.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/nameref.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/nameref.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/nameref.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/nameref.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/nameref.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/nameref.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/refcount/refcount.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/refcount/refcount.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/refcount/refcount.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/refcount/refcount.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/refcount/refcount.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/refcount/refcount.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/refcount/refcount.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/refcount/refcount.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/refcount/refcount.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/refcount/refcount.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/refcount/refcount.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/ltxcmds/ltxcmds.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/infwarerr/infwarerr.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/gettitlestring/gettitlestring.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/gettitlestring/gettitlestring.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/gettitlestring/gettitlestring.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/gettitlestring/gettitlestring.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/gettitlestring/gettitlestring.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/gettitlestring/gettitlestring.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/gettitlestring/gettitlestring.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/gettitlestring/gettitlestring.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/gettitlestring/gettitlestring.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/gettitlestring/gettitlestring.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/gettitlestring/gettitlestring.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/kvoptions/kvoptions.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/ltxcmds/ltxcmds.sty
INPUT ./paper.out
INPUT paper.out
INPUT ./paper.out
INPUT paper.out
INPUT ./paper.out
INPUT paper.out
INPUT ./paper.out
INPUT paper.out
OUTPUT paper.pdf
INPUT ./paper.out
INPUT ./paper.out
OUTPUT paper.out
INPUT /usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
INPUT /usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
INPUT /usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
INPUT /usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmcsc10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti8.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmbx10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmcsc10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam7.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti10.tfm
INPUT /usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map
INPUT ./fig_levels.png
INPUT ./fig_levels.png
INPUT fig_levels.png
INPUT ./fig_levels.png
INPUT ./fig_levels.png
INPUT ./fig_level_cycle.png
INPUT ./fig_level_cycle.png
INPUT fig_level_cycle.png
INPUT ./fig_level_cycle.png
INPUT ./fig_level_cycle.png
INPUT ./fig_edge_switch.png
INPUT ./fig_edge_switch.png
INPUT fig_edge_switch.png
INPUT ./fig_edge_switch.png
INPUT ./fig_edge_switch.png
INPUT ./fig_parity_subgraph.png
INPUT ./fig_parity_subgraph.png
INPUT fig_parity_subgraph.png
INPUT ./fig_parity_subgraph.png
INPUT ./fig_parity_subgraph.png
INPUT ./figures/n21_duals.png
INPUT ./figures/n21_duals.png
INPUT figures/n21_duals.png
INPUT ./figures/n21_duals.png
INPUT ./figures/n21_duals.png
INPUT paper.aux
INPUT ./paper.out
INPUT ./paper.out
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb
papers/even_level_graph_generators/paper.log
+145 -56
View File
@@ -1,12 +1,12 @@
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 22 MAY 2026 11:48 This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 22 MAY 2026 12:23
entering extended mode entering extended mode
restricted \write18 enabled. restricted \write18 enabled.
file:line:error style messages enabled.
%&-line parsing enabled. %&-line parsing enabled.
**/Users/didericis/Code/math-research/papers/even_level_graph_generators/paper.tex **paper.tex
(/Users/didericis/Code/math-research/papers/even_level_graph_generators/paper.tex (./paper.tex
LaTeX2e <2021-11-15> patch level 1 LaTeX2e <2021-11-15> patch level 1
L3 programming layer <2022-02-24> (/usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls L3 programming layer <2022-02-24>
(/usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
Document Class: amsart 2020/05/29 v2.20.6 Document Class: amsart 2020/05/29 v2.20.6
\linespacing=\dimen138 \linespacing=\dimen138
\normalparindent=\dimen139 \normalparindent=\dimen139
@@ -18,14 +18,17 @@ Package: amsmath 2021/10/15 v2.17l AMS math features
For additional information on amsmath, use the `?' option. For additional information on amsmath, use the `?' option.
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
Package: amstext 2021/08/26 v2.01 AMS text Package: amstext 2021/08/26 v2.01 AMS text
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty (/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
File: amsgen.sty 1999/11/30 v2.0 generic functions File: amsgen.sty 1999/11/30 v2.0 generic functions
\@emptytoks=\toks16 \@emptytoks=\toks16
\ex@=\dimen140 \ex@=\dimen140
)) (/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty ))
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
Package: amsbsy 1999/11/29 v1.2d Bold Symbols Package: amsbsy 1999/11/29 v1.2d Bold Symbols
\pmbraise@=\dimen141 \pmbraise@=\dimen141
) (/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty )
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
Package: amsopn 2021/08/26 v2.02 operator names Package: amsopn 2021/08/26 v2.02 operator names
) )
\inf@bad=\count185 \inf@bad=\count185
@@ -66,10 +69,13 @@ LaTeX Font Info: Redeclaring font encoding OMS on input line 744.
LaTeX Info: Redefining \[ on input line 2938. LaTeX Info: Redefining \[ on input line 2938.
LaTeX Info: Redefining \] on input line 2939. LaTeX Info: Redefining \] on input line 2939.
) )
LaTeX Font Info: Trying to load font information for U+msa on input line 397. LaTeX Font Info: Trying to load font information for U+msa on input line 397
.
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd (/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
File: umsa.fd 2013/01/14 v3.01 AMS symbols A File: umsa.fd 2013/01/14 v3.01 AMS symbols A
) (/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty )
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support
\symAMSa=\mathgroup4 \symAMSa=\mathgroup4
\symAMSb=\mathgroup5 \symAMSb=\mathgroup5
@@ -100,51 +106,69 @@ LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold'
\thm@postskip=\skip55 \thm@postskip=\skip55
\thm@headsep=\skip56 \thm@headsep=\skip56
\dth@everypar=\toks26 \dth@everypar=\toks26
) (/usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/hyperref.sty )
(/usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/hyperref.sty
Package: hyperref 2022-02-21 v7.00n Hypertext links for LaTeX Package: hyperref 2022-02-21 v7.00n Hypertext links for LaTeX
(/usr/local/texlive/2022/texmf-dist/tex/generic/ltxcmds/ltxcmds.sty (/usr/local/texlive/2022/texmf-dist/tex/generic/ltxcmds/ltxcmds.sty
Package: ltxcmds 2020-05-10 v1.25 LaTeX kernel commands for general use (HO) Package: ltxcmds 2020-05-10 v1.25 LaTeX kernel commands for general use (HO)
) (/usr/local/texlive/2022/texmf-dist/tex/generic/iftex/iftex.sty )
(/usr/local/texlive/2022/texmf-dist/tex/generic/iftex/iftex.sty
Package: iftex 2022/02/03 v1.0f TeX engine tests Package: iftex 2022/02/03 v1.0f TeX engine tests
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pdftexcmds/pdftexcmds.sty )
Package: pdftexcmds 2020-06-27 v0.33 Utility functions of pdfTeX for LuaTeX (HO) (/usr/local/texlive/2022/texmf-dist/tex/generic/pdftexcmds/pdftexcmds.sty
Package: pdftexcmds 2020-06-27 v0.33 Utility functions of pdfTeX for LuaTeX (HO
)
(/usr/local/texlive/2022/texmf-dist/tex/generic/infwarerr/infwarerr.sty (/usr/local/texlive/2022/texmf-dist/tex/generic/infwarerr/infwarerr.sty
Package: infwarerr 2019/12/03 v1.5 Providing info/warning/error messages (HO) Package: infwarerr 2019/12/03 v1.5 Providing info/warning/error messages (HO)
) )
Package pdftexcmds Info: \pdf@primitive is available. Package pdftexcmds Info: \pdf@primitive is available.
Package pdftexcmds Info: \pdf@ifprimitive is available. Package pdftexcmds Info: \pdf@ifprimitive is available.
Package pdftexcmds Info: \pdfdraftmode found. Package pdftexcmds Info: \pdfdraftmode found.
) (/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty )
(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
Package: keyval 2014/10/28 v1.15 key=value parser (DPC) Package: keyval 2014/10/28 v1.15 key=value parser (DPC)
\KV@toks@=\toks27 \KV@toks@=\toks27
) (/usr/local/texlive/2022/texmf-dist/tex/generic/kvsetkeys/kvsetkeys.sty )
(/usr/local/texlive/2022/texmf-dist/tex/generic/kvsetkeys/kvsetkeys.sty
Package: kvsetkeys 2019/12/15 v1.18 Key value parser (HO) Package: kvsetkeys 2019/12/15 v1.18 Key value parser (HO)
) (/usr/local/texlive/2022/texmf-dist/tex/generic/kvdefinekeys/kvdefinekeys.sty )
(/usr/local/texlive/2022/texmf-dist/tex/generic/kvdefinekeys/kvdefinekeys.sty
Package: kvdefinekeys 2019-12-19 v1.6 Define keys (HO) Package: kvdefinekeys 2019-12-19 v1.6 Define keys (HO)
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pdfescape/pdfescape.sty )
(/usr/local/texlive/2022/texmf-dist/tex/generic/pdfescape/pdfescape.sty
Package: pdfescape 2019/12/09 v1.15 Implements pdfTeX's escape features (HO) Package: pdfescape 2019/12/09 v1.15 Implements pdfTeX's escape features (HO)
) (/usr/local/texlive/2022/texmf-dist/tex/latex/hycolor/hycolor.sty )
(/usr/local/texlive/2022/texmf-dist/tex/latex/hycolor/hycolor.sty
Package: hycolor 2020-01-27 v1.10 Color options for hyperref/bookmark (HO) Package: hycolor 2020-01-27 v1.10 Color options for hyperref/bookmark (HO)
) (/usr/local/texlive/2022/texmf-dist/tex/latex/letltxmacro/letltxmacro.sty )
(/usr/local/texlive/2022/texmf-dist/tex/latex/letltxmacro/letltxmacro.sty
Package: letltxmacro 2019/12/03 v1.6 Let assignment for LaTeX macros (HO) Package: letltxmacro 2019/12/03 v1.6 Let assignment for LaTeX macros (HO)
) (/usr/local/texlive/2022/texmf-dist/tex/latex/auxhook/auxhook.sty )
(/usr/local/texlive/2022/texmf-dist/tex/latex/auxhook/auxhook.sty
Package: auxhook 2019-12-17 v1.6 Hooks for auxiliary files (HO) Package: auxhook 2019-12-17 v1.6 Hooks for auxiliary files (HO)
) (/usr/local/texlive/2022/texmf-dist/tex/latex/kvoptions/kvoptions.sty )
(/usr/local/texlive/2022/texmf-dist/tex/latex/kvoptions/kvoptions.sty
Package: kvoptions 2020-10-07 v3.14 Key value format for package options (HO) Package: kvoptions 2020-10-07 v3.14 Key value format for package options (HO)
) )
\@linkdim=\dimen150 \@linkdim=\dimen150
\Hy@linkcounter=\count272 \Hy@linkcounter=\count272
\Hy@pagecounter=\count273 \Hy@pagecounter=\count273
(/usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/pd1enc.def (/usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/pd1enc.def
File: pd1enc.def 2022-02-21 v7.00n Hyperref: PDFDocEncoding definition (HO) File: pd1enc.def 2022-02-21 v7.00n Hyperref: PDFDocEncoding definition (HO)
Now handling font encoding PD1 ... Now handling font encoding PD1 ...
... no UTF-8 mapping file for font encoding PD1 ... no UTF-8 mapping file for font encoding PD1
) (/usr/local/texlive/2022/texmf-dist/tex/generic/intcalc/intcalc.sty )
(/usr/local/texlive/2022/texmf-dist/tex/generic/intcalc/intcalc.sty
Package: intcalc 2019/12/15 v1.3 Expandable calculations with integers (HO) Package: intcalc 2019/12/15 v1.3 Expandable calculations with integers (HO)
) (/usr/local/texlive/2022/texmf-dist/tex/generic/etexcmds/etexcmds.sty )
(/usr/local/texlive/2022/texmf-dist/tex/generic/etexcmds/etexcmds.sty
Package: etexcmds 2019/12/15 v1.7 Avoid name clashes with e-TeX commands (HO) Package: etexcmds 2019/12/15 v1.7 Avoid name clashes with e-TeX commands (HO)
) )
\Hy@SavedSpaceFactor=\count274 \Hy@SavedSpaceFactor=\count274
(/usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/puenc.def (/usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/puenc.def
File: puenc.def 2022-02-21 v7.00n Hyperref: PDF Unicode definition (HO) File: puenc.def 2022-02-21 v7.00n Hyperref: PDF Unicode definition (HO)
Now handling font encoding PU ... Now handling font encoding PU ...
@@ -158,16 +182,20 @@ Package hyperref Info: Backreferencing OFF on input line 4157.
Package hyperref Info: Implicit mode ON; LaTeX internals redefined. Package hyperref Info: Implicit mode ON; LaTeX internals redefined.
Package hyperref Info: Bookmarks ON on input line 4390. Package hyperref Info: Bookmarks ON on input line 4390.
\c@Hy@tempcnt=\count275 \c@Hy@tempcnt=\count275
(/usr/local/texlive/2022/texmf-dist/tex/latex/url/url.sty (/usr/local/texlive/2022/texmf-dist/tex/latex/url/url.sty
\Urlmuskip=\muskip17 \Urlmuskip=\muskip17
Package: url 2013/09/16 ver 3.4 Verb mode for urls, etc. Package: url 2013/09/16 ver 3.4 Verb mode for urls, etc.
) )
LaTeX Info: Redefining \url on input line 4749. LaTeX Info: Redefining \url on input line 4749.
\XeTeXLinkMargin=\dimen151 \XeTeXLinkMargin=\dimen151
(/usr/local/texlive/2022/texmf-dist/tex/generic/bitset/bitset.sty (/usr/local/texlive/2022/texmf-dist/tex/generic/bitset/bitset.sty
Package: bitset 2019/12/09 v1.3 Handle bit-vector datatype (HO) Package: bitset 2019/12/09 v1.3 Handle bit-vector datatype (HO)
(/usr/local/texlive/2022/texmf-dist/tex/generic/bigintcalc/bigintcalc.sty (/usr/local/texlive/2022/texmf-dist/tex/generic/bigintcalc/bigintcalc.sty
Package: bigintcalc 2019/12/15 v1.5 Expandable calculations on big integers (HO) Package: bigintcalc 2019/12/15 v1.5 Expandable calculations on big integers (HO
)
)) ))
\Fld@menulength=\count276 \Fld@menulength=\count276
\Field@Width=\dimen152 \Field@Width=\dimen152
@@ -181,6 +209,7 @@ Package hyperref Info: Link coloring with OCG OFF on input line 6052.
Package hyperref Info: PDF/A mode OFF on input line 6057. Package hyperref Info: PDF/A mode OFF on input line 6057.
LaTeX Info: Redefining \ref on input line 6097. LaTeX Info: Redefining \ref on input line 6097.
LaTeX Info: Redefining \pageref on input line 6101. LaTeX Info: Redefining \pageref on input line 6101.
(/usr/local/texlive/2022/texmf-dist/tex/latex/base/atbegshi-ltx.sty (/usr/local/texlive/2022/texmf-dist/tex/latex/base/atbegshi-ltx.sty
Package: atbegshi-ltx 2021/01/10 v1.0c Emulation of the original atbegshi Package: atbegshi-ltx 2021/01/10 v1.0c Emulation of the original atbegshi
package with kernel methods package with kernel methods
@@ -190,20 +219,26 @@ package with kernel methods
\c@Hfootnote=\count279 \c@Hfootnote=\count279
) )
Package hyperref Info: Driver (autodetected): hpdftex. Package hyperref Info: Driver (autodetected): hpdftex.
(/usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/hpdftex.def (/usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/hpdftex.def
File: hpdftex.def 2022-02-21 v7.00n Hyperref driver for pdfTeX File: hpdftex.def 2022-02-21 v7.00n Hyperref driver for pdfTeX
(/usr/local/texlive/2022/texmf-dist/tex/latex/base/atveryend-ltx.sty (/usr/local/texlive/2022/texmf-dist/tex/latex/base/atveryend-ltx.sty
Package: atveryend-ltx 2020/08/19 v1.0a Emulation of the original atveryend package Package: atveryend-ltx 2020/08/19 v1.0a Emulation of the original atveryend pac
kage
with kernel methods with kernel methods
) )
\Fld@listcount=\count280 \Fld@listcount=\count280
\c@bookmark@seq@number=\count281 \c@bookmark@seq@number=\count281
(/usr/local/texlive/2022/texmf-dist/tex/latex/rerunfilecheck/rerunfilecheck.sty (/usr/local/texlive/2022/texmf-dist/tex/latex/rerunfilecheck/rerunfilecheck.sty
Package: rerunfilecheck 2019/12/05 v1.9 Rerun checks for auxiliary files (HO) Package: rerunfilecheck 2019/12/05 v1.9 Rerun checks for auxiliary files (HO)
(/usr/local/texlive/2022/texmf-dist/tex/generic/uniquecounter/uniquecounter.sty (/usr/local/texlive/2022/texmf-dist/tex/generic/uniquecounter/uniquecounter.sty
Package: uniquecounter 2019/12/15 v1.4 Provide unlimited unique counter (HO) Package: uniquecounter 2019/12/15 v1.4 Provide unlimited unique counter (HO)
) )
Package uniquecounter Info: New unique counter `rerunfilecheck' on input line 286. Package uniquecounter Info: New unique counter `rerunfilecheck' on input line 2
86.
) )
\Hy@SectionHShift=\skip57 \Hy@SectionHShift=\skip57
) (/usr/local/texlive/2022/texmf-dist/tex/latex/enumitem/enumitem.sty ) (/usr/local/texlive/2022/texmf-dist/tex/latex/enumitem/enumitem.sty
@@ -214,16 +249,21 @@ Package: enumitem 2019/06/20 v3.9 Customized lists
\enit@inbox=\box53 \enit@inbox=\box53
\enit@count@id=\count282 \enit@count@id=\count282
\enitdp@description=\count283 \enitdp@description=\count283
) (/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty )
(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
Package: graphicx 2021/09/16 v1.2d Enhanced LaTeX Graphics (DPC,SPQR) Package: graphicx 2021/09/16 v1.2d Enhanced LaTeX Graphics (DPC,SPQR)
(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty (/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
Package: graphics 2021/03/04 v1.4d Standard LaTeX Graphics (DPC,SPQR) Package: graphics 2021/03/04 v1.4d Standard LaTeX Graphics (DPC,SPQR)
(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty (/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
Package: trig 2021/08/11 v1.11 sin cos tan (DPC) Package: trig 2021/08/11 v1.11 sin cos tan (DPC)
) (/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg )
(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg
File: graphics.cfg 2016/06/04 v1.11 sample graphics configuration File: graphics.cfg 2016/06/04 v1.11 sample graphics configuration
) )
Package graphics Info: Driver file: pdftex.def on input line 107. Package graphics Info: Driver file: pdftex.def on input line 107.
(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def (/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def
File: pdftex.def 2020/10/05 v1.2a Graphics/color driver for pdftex File: pdftex.def 2020/10/05 v1.2a Graphics/color driver for pdftex
)) ))
@@ -231,11 +271,13 @@ File: pdftex.def 2020/10/05 v1.2a Graphics/color driver for pdftex
\Gin@req@width=\dimen156 \Gin@req@width=\dimen156
) )
\c@theorem=\count284 \c@theorem=\count284
(/usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def (/usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
File: l3backend-pdftex.def 2022-02-07 L3 backend support: PDF output (pdfTeX) File: l3backend-pdftex.def 2022-02-07 L3 backend support: PDF output (pdfTeX)
\l__color_backend_stack_int=\count285 \l__color_backend_stack_int=\count285
\l__pdf_internal_box=\box54 \l__pdf_internal_box=\box54
) (./paper.aux) )
(./paper.aux)
\openout1 = `paper.aux'. \openout1 = `paper.aux'.
LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 60. LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 60.
@@ -257,19 +299,26 @@ LaTeX Font Info: ... okay on input line 60.
LaTeX Font Info: Checking defaults for PU/pdf/m/n on input line 60. LaTeX Font Info: Checking defaults for PU/pdf/m/n on input line 60.
LaTeX Font Info: ... okay on input line 60. LaTeX Font Info: ... okay on input line 60.
LaTeX Font Info: Trying to load font information for U+msa on input line 60. LaTeX Font Info: Trying to load font information for U+msa on input line 60.
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd (/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
File: umsa.fd 2013/01/14 v3.01 AMS symbols A File: umsa.fd 2013/01/14 v3.01 AMS symbols A
) )
LaTeX Font Info: Trying to load font information for U+msb on input line 60. LaTeX Font Info: Trying to load font information for U+msb on input line 60.
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd (/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd
File: umsb.fd 2013/01/14 v3.01 AMS symbols B File: umsb.fd 2013/01/14 v3.01 AMS symbols B
) )
Package hyperref Info: Link coloring OFF on input line 60. Package hyperref Info: Link coloring OFF on input line 60.
(/usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/nameref.sty (/usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/nameref.sty
Package: nameref 2021-04-02 v2.47 Cross-referencing by name of section Package: nameref 2021-04-02 v2.47 Cross-referencing by name of section
(/usr/local/texlive/2022/texmf-dist/tex/latex/refcount/refcount.sty (/usr/local/texlive/2022/texmf-dist/tex/latex/refcount/refcount.sty
Package: refcount 2019/12/15 v3.6 Data extraction from label references (HO) Package: refcount 2019/12/15 v3.6 Data extraction from label references (HO)
) (/usr/local/texlive/2022/texmf-dist/tex/generic/gettitlestring/gettitlestring.sty )
(/usr/local/texlive/2022/texmf-dist/tex/generic/gettitlestring/gettitlestring.s
ty
Package: gettitlestring 2019/12/15 v1.6 Cleanup title references (HO) Package: gettitlestring 2019/12/15 v1.6 Cleanup title references (HO)
) )
\c@section@level=\count286 \c@section@level=\count286
@@ -281,6 +330,7 @@ LaTeX Info: Redefining \nameref on input line 60.
\@outlinefile=\write3 \@outlinefile=\write3
\openout3 = `paper.out'. \openout3 = `paper.out'.
(/usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii (/usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
[Loading MPS to PDF converter (version 2006.09.02).] [Loading MPS to PDF converter (version 2006.09.02).]
\scratchcounter=\count287 \scratchcounter=\count287
@@ -296,11 +346,15 @@ LaTeX Info: Redefining \nameref on input line 60.
\everyMPtoPDFconversion=\toks30 \everyMPtoPDFconversion=\toks30
) (/usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty ) (/usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
Package: epstopdf-base 2020-01-24 v2.11 Base part for package epstopdf Package: epstopdf-base 2020-01-24 v2.11 Base part for package epstopdf
Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 485. Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 4
85.
(/usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg (/usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg
File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Live File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv
)) [1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] e
<fig_levels.png, id=52, 454.21695pt x 391.34206pt> ))
[1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}]
<fig_levels.png, id=56, 454.21695pt x 391.34206pt>
File: fig_levels.png Graphic file (type png) File: fig_levels.png Graphic file (type png)
<use fig_levels.png> <use fig_levels.png>
Package pdftex.def Info: fig_levels.png used on input line 184. Package pdftex.def Info: fig_levels.png used on input line 184.
@@ -309,7 +363,7 @@ Package pdftex.def Info: fig_levels.png used on input line 184.
LaTeX Warning: `h' float specifier changed to `ht'. LaTeX Warning: `h' float specifier changed to `ht'.
<fig_level_cycle.png, id=54, 452.04884pt x 391.34206pt> <fig_level_cycle.png, id=58, 452.04884pt x 391.34206pt>
File: fig_level_cycle.png Graphic file (type png) File: fig_level_cycle.png Graphic file (type png)
<use fig_level_cycle.png> <use fig_level_cycle.png>
Package pdftex.def Info: fig_level_cycle.png used on input line 198. Package pdftex.def Info: fig_level_cycle.png used on input line 198.
@@ -317,7 +371,7 @@ Package pdftex.def Info: fig_level_cycle.png used on input line 198.
LaTeX Warning: `h' float specifier changed to `ht'. LaTeX Warning: `h' float specifier changed to `ht'.
<fig_edge_switch.png, id=56, 859.65166pt x 378.33345pt> <fig_edge_switch.png, id=60, 859.65166pt x 378.33345pt>
File: fig_edge_switch.png Graphic file (type png) File: fig_edge_switch.png Graphic file (type png)
<use fig_edge_switch.png> <use fig_edge_switch.png>
Package pdftex.def Info: fig_edge_switch.png used on input line 217. Package pdftex.def Info: fig_edge_switch.png used on input line 217.
@@ -325,7 +379,7 @@ Package pdftex.def Info: fig_edge_switch.png used on input line 217.
LaTeX Warning: `h' float specifier changed to `ht'. LaTeX Warning: `h' float specifier changed to `ht'.
<fig_parity_subgraph.png, id=58, 1076.46165pt x 319.79475pt> <fig_parity_subgraph.png, id=62, 1076.46165pt x 319.79475pt>
File: fig_parity_subgraph.png Graphic file (type png) File: fig_parity_subgraph.png Graphic file (type png)
<use fig_parity_subgraph.png> <use fig_parity_subgraph.png>
Package pdftex.def Info: fig_parity_subgraph.png used on input line 235. Package pdftex.def Info: fig_parity_subgraph.png used on input line 235.
@@ -333,38 +387,73 @@ Package pdftex.def Info: fig_parity_subgraph.png used on input line 235.
LaTeX Warning: `h' float specifier changed to `ht'. LaTeX Warning: `h' float specifier changed to `ht'.
[2] [3 <./fig_levels.png> <./fig_level_cycle.png>] [4 <./fig_edge_switch.png> <./fig_parity_subgraph.png>] [5] [2] [3 <./fig_levels.png> <./fig_level_cycle.png>] [4 <./fig_edge_switch.png> <
./fig_parity_subgraph.png>]
Package hyperref Warning: Token not allowed in a PDF string (Unicode): Package hyperref Warning: Token not allowed in a PDF string (Unicode):
(hyperref) removing `math shift' on input line 415. (hyperref) removing `math shift' on input line 315.
Package hyperref Warning: Token not allowed in a PDF string (Unicode): Package hyperref Warning: Token not allowed in a PDF string (Unicode):
(hyperref) removing `math shift' on input line 415. (hyperref) removing `math shift' on input line 315.
[6]
<figures/n21_duals.png, id=114, 1373.13pt x 867.24pt> Overfull \hbox (5.73714pt too wide) in paragraph at lines 327--331
\OT1/cmr/m/n/10 where $\OML/cmm/m/it/10 s\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 G\OT
1/cmr/m/n/10 )$ is the num-ber of valid sources of $\OML/cmm/m/it/10 G$\OT1/cmr
/m/n/10 . The flag-rooting is the automorphism-
[]
Underfull \hbox (badness 1112) in paragraph at lines 354--354
\OT1/cmr/m/n/10 mor-phism; \OT1/cmr/m/it/10 flag-rooted ELGs \OT1/cmr/m/n/10 is
the automorphism-free count
[]
[5] [6]
Package hyperref Warning: Token not allowed in a PDF string (Unicode):
(hyperref) removing `math shift' on input line 460.
Package hyperref Warning: Token not allowed in a PDF string (Unicode):
(hyperref) removing `math shift' on input line 460.
<figures/n21_duals.png, id=119, 1373.13pt x 867.24pt>
File: figures/n21_duals.png Graphic file (type png) File: figures/n21_duals.png Graphic file (type png)
<use figures/n21_duals.png> <use figures/n21_duals.png>
Package pdftex.def Info: figures/n21_duals.png used on input line 475. Package pdftex.def Info: figures/n21_duals.png used on input line 520.
(pdftex.def) Requested size: 360.0pt x 227.35617pt. (pdftex.def) Requested size: 360.0pt x 227.35617pt.
[7 <./figures/n21_duals.png>] [8] (./paper.aux) [7] [8 <./figures/n21_duals.png>] (./paper.aux)
Package rerunfilecheck Info: File `paper.out' has not changed. Package rerunfilecheck Info: File `paper.out' has not changed.
(rerunfilecheck) Checksum: 06539E3751DA0B503943CA6640B71438;911. (rerunfilecheck) Checksum: CF07A31709BA02BE3BA2BC89322768D0;1088.
) )
Here is how much of TeX's memory you used: Here is how much of TeX's memory you used:
9755 strings out of 478268 9758 strings out of 478268
151243 string characters out of 5846347 151077 string characters out of 5846347
454766 words of memory out of 5000000 455062 words of memory out of 5000000
27654 multiletter control sequences out of 15000+600000 27656 multiletter control sequences out of 15000+600000
475666 words of font info for 53 fonts, out of 8000000 for 9000 475666 words of font info for 53 fonts, out of 8000000 for 9000
1302 hyphenation exceptions out of 8191 1302 hyphenation exceptions out of 8191
69i,9n,76p,852b,504s stack positions out of 10000i,1000n,20000p,200000b,200000s 69i,9n,76p,781b,434s stack positions out of 10000i,1000n,20000p,200000b,200000s
</usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb> </usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb
Output written on paper.pdf (8 pages, 1093752 bytes). ></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb
></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb>
</usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb><
/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></u
sr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.pfb></usr
/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/l
ocal/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/loca
l/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb></usr/local/t
exlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb></usr/local/te
xlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></usr/local/texl
ive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/local/texli
ve/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb></usr/local/texlive
/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb>
Output written on paper.pdf (8 pages, 1098280 bytes).
PDF statistics: PDF statistics:
196 PDF objects out of 1000 (max. 8388607) 205 PDF objects out of 1000 (max. 8388607)
146 compressed objects within 2 object streams 155 compressed objects within 2 object streams
36 named destinations out of 1000 (max. 500000) 38 named destinations out of 1000 (max. 500000)
74 words of extra memory for PDF output out of 10000 (max. 10000000) 82 words of extra memory for PDF output out of 10000 (max. 10000000)
papers/even_level_graph_generators/paper.out
+3 -2
View File
@@ -2,5 +2,6 @@
\BOOKMARK [1][-]{section.2}{\376\377\0002\000.\000\040\000D\000e\000f\000i\000n\000i\000t\000i\000o\000n\000s}{}% 2 \BOOKMARK [1][-]{section.2}{\376\377\0002\000.\000\040\000D\000e\000f\000i\000n\000i\000t\000i\000o\000n\000s}{}% 2
\BOOKMARK [1][-]{section.3}{\376\377\0003\000.\000\040\000O\000u\000t\000e\000r\000p\000l\000a\000n\000a\000r\000i\000t\000y\000\040\000o\000f\000\040\000l\000e\000v\000e\000l\000\040\000c\000o\000m\000p\000o\000n\000e\000n\000t\000s}{}% 3 \BOOKMARK [1][-]{section.3}{\376\377\0003\000.\000\040\000O\000u\000t\000e\000r\000p\000l\000a\000n\000a\000r\000i\000t\000y\000\040\000o\000f\000\040\000l\000e\000v\000e\000l\000\040\000c\000o\000m\000p\000o\000n\000e\000n\000t\000s}{}% 3
\BOOKMARK [1][-]{section.4}{\376\377\0004\000.\000\040\000E\000v\000e\000n\000\040\000L\000e\000v\000e\000l\000\040\000G\000r\000a\000p\000h\000s}{}% 4 \BOOKMARK [1][-]{section.4}{\376\377\0004\000.\000\040\000E\000v\000e\000n\000\040\000L\000e\000v\000e\000l\000\040\000G\000r\000a\000p\000h\000s}{}% 4
\BOOKMARK [2][-]{section*.1}{\376\377\000T\000h\000e\000\040\000b\000o\000u\000n\000d\000a\000r\000y\000\040\000c\000a\000s\000e\000\040\000n\000\040\000=\000\040\0002\0001}{section.4}% 5 \BOOKMARK [2][-]{section*.1}{\376\377\000E\000n\000u\000m\000e\000r\000a\000t\000i\000o\000n\000\040\000f\000o\000r\000\040\000s\000m\000a\000l\000l\000\040\000n}{section.4}% 5
\BOOKMARK [1][-]{section*.2}{\376\377\000R\000e\000f\000e\000r\000e\000n\000c\000e\000s}{}% 6 \BOOKMARK [2][-]{section*.2}{\376\377\000T\000h\000e\000\040\000b\000o\000u\000n\000d\000a\000r\000y\000\040\000c\000a\000s\000e\000\040\000n\000\040\000=\000\040\0002\0001}{section.4}% 6
\BOOKMARK [1][-]{section*.3}{\376\377\000R\000e\000f\000e\000r\000e\000n\000c\000e\000s}{}% 7
papers/even_level_graph_generators/paper.pdf
BIN
View File
Binary file not shown.
papers/even_level_graph_generators/paper.tex
+45
View File
@@ -310,6 +310,51 @@ $\{\text{yellow}, \text{green}\}$. The combined assignment is a
proper $4$-coloring of $G$. proper $4$-coloring of $G$.
\end{proof} \end{proof}
\subsection*{Enumeration for small $n$}
Even Level Graphs are scarce among triangulations. For each $n$ we
enumerated all iso classes of plane triangulations and tested, for every
choice of source vertex, whether all level subgraphs are bipartite
(equivalently, whether every level cycle is even).
Table~\ref{tab:elg-counts} records, for $4 \leq n \leq 11$: the number of
triangulation iso classes; how many of them admit at least one Even Level
Graph source; the number of Even Level Graph iso classes (pairs $(G, S)$
up to isomorphism, i.e.\ valid sources counted up to $\mathrm{Aut}(G)$);
and the number of \emph{flag-rooted} Even Level Graphs,
\[
\sum_{G} \frac{4E}{|\mathrm{Aut}(G)|}\, s(G),
\qquad E = 3n - 6,
\]
where $s(G)$ is the number of valid sources of $G$. The flag-rooting is the
automorphism-free count: $\mathrm{Aut}(G)$ acts freely on the $4E$ flags of a
$3$-connected triangulation, so every summand is an integer.
The smallest Even Level Graph is the octahedron at $n = 6$: from any vertex
the four neighbours form a $4$-cycle at level $1$ and the antipode sits alone
at level $2$. Below $n = 6$ every triangulation forces an odd level cycle, so
no Even Level Graph exists.
\begin{table}[ht]
\centering
\begin{tabular}{ccccc}
$n$ & triangulations & with ELG source & ELG iso classes & flag-rooted ELGs \\\hline
$4$ & $1$ & $0$ & $0$ & $0$ \\
$5$ & $1$ & $0$ & $0$ & $0$ \\
$6$ & $2$ & $1$ & $1$ & $6$ \\
$7$ & $5$ & $2$ & $2$ & $45$ \\
$8$ & $14$ & $5$ & $6$ & $186$ \\
$9$ & $50$ & $13$ & $14$ & $651$ \\
$10$ & $233$ & $37$ & $45$ & $2766$ \\
$11$ & $1249$ & $129$ & $169$ & $14346$ \\
\end{tabular}
\caption{Even Level Graph counts for $4 \leq n \leq 11$. The
\emph{triangulations} column is the number of plane-triangulation iso classes
(OEIS A000109). \emph{ELG iso classes} counts pairs $(G, S)$ up to
isomorphism; \emph{flag-rooted ELGs} is the automorphism-free count
$\sum_G \tfrac{4E}{|\mathrm{Aut}(G)|}\,s(G)$.}
\label{tab:elg-counts}
\end{table}
\begin{definition}[Derived level graph] \begin{definition}[Derived level graph]
\label{def:derived-level-graph} \label{def:derived-level-graph}
Let $G$ be an Even Level Graph with level source $S$, and let $E$ and Let $G$ be an Even Level Graph with level source $S$, and let $E$ and