Add diamond coloring conjecture, parity-separation reformulation, and counterexample search
Extends paper with: a notation section for color-class preimages; the plane diamond coloring definition (4-coloring whose two classes lift to a 2-coloring of some BFS-rooted diamond scaffold); a connectedness lemma for the scaffold; a proposition reformulating the property as parity- separation of two color classes by BFS layers; a remark noting this is strictly stronger than 4CT; and the conjecture that every maximal planar graph admits such a coloring. Adds plane_diamond_coloring.py with get_plane_diamond_scaffold and a counterexample search that reduces the per-root check to 4-colorability of an auxiliary graph forcing two colors onto opposite parity layers. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
@@ -1,9 +1,15 @@
|
||||
\relax
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{Notation}}{1}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Definitions}}{1}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Results}}{1}{}\protected@file@percent }
|
||||
\citation{appel1977every}
|
||||
\citation{robertson1997four}
|
||||
\bibcite{appel1977every}{1}
|
||||
\bibcite{robertson1997four}{2}
|
||||
\newlabel{tocindent-1}{0pt}
|
||||
\newlabel{tocindent0}{0pt}
|
||||
\newlabel{tocindent0}{12.7778pt}
|
||||
\newlabel{tocindent1}{17.77782pt}
|
||||
\newlabel{tocindent2}{0pt}
|
||||
\newlabel{tocindent3}{0pt}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Definitions}}{1}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Results}}{1}{}\protected@file@percent }
|
||||
\gdef \@abspage@last{1}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{2}{}\protected@file@percent }
|
||||
\gdef \@abspage@last{2}
|
||||
|
||||
@@ -1,5 +1,5 @@
|
||||
# Fdb version 3
|
||||
["pdflatex"] 1778341899 "paper.tex" "paper.pdf" "paper" 1778341900
|
||||
["pdflatex"] 1778344156 "paper.tex" "paper.pdf" "paper" 1778344156
|
||||
"/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 ""
|
||||
@@ -18,9 +18,11 @@
|
||||
"/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/cmti7.tfm" 1136768653 1492 86331993fe614793f5e7e755835c31c5 ""
|
||||
"/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/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/cmmi10.pfb" 1248133631 36299 5f9df58c2139e7edcf37c8fca4bd384d ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb" 1248133631 36281 c355509802a035cadc5f15869451dcee ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb" 1248133631 35752 024fb6c41858982481f6968b5fc26508 ""
|
||||
@@ -29,6 +31,7 @@
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb" 1248133631 32569 5e5ddc8df908dea60932f3c484a54c0d ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb" 1248133631 32716 08e384dc442464e7285e891af9f45947 ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb" 1248133631 37944 359e864bd06cde3b1cf57bb20757fb06 ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti7.pfb" 1248133631 36607 d654cb3f2bc54f57509240071db3bffa ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb" 1248133631 35660 fb24af7afbadb71801619f1415838111 ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb" 1248133631 31764 459c573c03a4949a528c2cc7f557e217 ""
|
||||
"/usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls" 1591045760 61881 a7369c346c2922a758ae6283cc1ed014 ""
|
||||
@@ -45,8 +48,8 @@
|
||||
"/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.cnf" 1647878952 577 209b46be99c9075fd74d4c0369380e8c ""
|
||||
"paper.aux" 1778341900 377 e4a4bae3d140fd6461ef3b90b48a4e36 "pdflatex"
|
||||
"paper.tex" 1778341891 3894 9067f9f68588b27557ec016c8a9ca5e1 ""
|
||||
"paper.aux" 1778344156 697 3f22ac2880dcfce06793ac851e5b5d36 "pdflatex"
|
||||
"paper.tex" 1778344142 7304 251fb4f84f05f1ed3480fd521a52c92f ""
|
||||
(generated)
|
||||
"paper.aux"
|
||||
"paper.log"
|
||||
|
||||
@@ -144,9 +144,12 @@ INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm7
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti10.tfm
|
||||
OUTPUT paper.pdf
|
||||
INPUT /usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti7.tfm
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti7.tfm
|
||||
INPUT paper.aux
|
||||
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/cmr10.pfb
|
||||
@@ -155,5 +158,6 @@ 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/cmti7.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
|
||||
|
||||
@@ -1,4 +1,4 @@
|
||||
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 9 MAY 2026 11:51
|
||||
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 9 MAY 2026 12:29
|
||||
entering extended mode
|
||||
restricted \write18 enabled.
|
||||
%&-line parsing enabled.
|
||||
@@ -117,56 +117,58 @@ File: l3backend-pdftex.def 2022-02-07 L3 backend support: PDF output (pdfTeX)
|
||||
(./paper.aux)
|
||||
\openout1 = `paper.aux'.
|
||||
|
||||
LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 52.
|
||||
LaTeX Font Info: ... okay on input line 52.
|
||||
LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 52.
|
||||
LaTeX Font Info: ... okay on input line 52.
|
||||
LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 52.
|
||||
LaTeX Font Info: ... okay on input line 52.
|
||||
LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 52.
|
||||
LaTeX Font Info: ... okay on input line 52.
|
||||
LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 52.
|
||||
LaTeX Font Info: ... okay on input line 52.
|
||||
LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 52.
|
||||
LaTeX Font Info: ... okay on input line 52.
|
||||
LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 52.
|
||||
LaTeX Font Info: ... okay on input line 52.
|
||||
LaTeX Font Info: Trying to load font information for U+msa on input line 52.
|
||||
LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 54.
|
||||
LaTeX Font Info: ... okay on input line 54.
|
||||
LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 54.
|
||||
LaTeX Font Info: ... okay on input line 54.
|
||||
LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 54.
|
||||
LaTeX Font Info: ... okay on input line 54.
|
||||
LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 54.
|
||||
LaTeX Font Info: ... okay on input line 54.
|
||||
LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 54.
|
||||
LaTeX Font Info: ... okay on input line 54.
|
||||
LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 54.
|
||||
LaTeX Font Info: ... okay on input line 54.
|
||||
LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 54.
|
||||
LaTeX Font Info: ... okay on input line 54.
|
||||
LaTeX Font Info: Trying to load font information for U+msa on input line 54.
|
||||
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
|
||||
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 52.
|
||||
LaTeX Font Info: Trying to load font information for U+msb on input line 54.
|
||||
|
||||
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd
|
||||
File: umsb.fd 2013/01/14 v3.01 AMS symbols B
|
||||
) [1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] (./
|
||||
paper.aux) )
|
||||
) [1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] [2]
|
||||
(./paper.aux) )
|
||||
Here is how much of TeX's memory you used:
|
||||
1714 strings out of 478268
|
||||
23914 string characters out of 5846347
|
||||
323202 words of memory out of 5000000
|
||||
19800 multiletter control sequences out of 15000+600000
|
||||
475666 words of font info for 53 fonts, out of 8000000 for 9000
|
||||
1725 strings out of 478268
|
||||
24057 string characters out of 5846347
|
||||
325269 words of memory out of 5000000
|
||||
19810 multiletter control sequences out of 15000+600000
|
||||
476364 words of font info for 55 fonts, out of 8000000 for 9000
|
||||
1302 hyphenation exceptions out of 8191
|
||||
69i,5n,76p,381b,204s 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/c
|
||||
mcsc10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/c
|
||||
mmi10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cm
|
||||
mi7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr1
|
||||
0.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.p
|
||||
fb></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></u
|
||||
sr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></us
|
||||
r/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>
|
||||
Output written on paper.pdf (1 page, 139867 bytes).
|
||||
69i,6n,76p,613b,223s 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/c
|
||||
m/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/c
|
||||
mr10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr
|
||||
7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.p
|
||||
fb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pf
|
||||
b></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/cmti7.pfb></u
|
||||
sr/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>
|
||||
Output written on paper.pdf (2 pages, 176410 bytes).
|
||||
PDF statistics:
|
||||
68 PDF objects out of 1000 (max. 8388607)
|
||||
40 compressed objects within 1 object stream
|
||||
81 PDF objects out of 1000 (max. 8388607)
|
||||
48 compressed objects within 1 object stream
|
||||
0 named destinations out of 1000 (max. 500000)
|
||||
1 words of extra memory for PDF output out of 10000 (max. 10000000)
|
||||
|
||||
|
||||
Binary file not shown.
@@ -38,6 +38,8 @@
|
||||
|
||||
\newtheorem{theorem}{Theorem}[section]
|
||||
\newtheorem{lemma}[theorem]{Lemma}
|
||||
\newtheorem{proposition}[theorem]{Proposition}
|
||||
\newtheorem{conjecture}[theorem]{Conjecture}
|
||||
|
||||
\theoremstyle{definition}
|
||||
\newtheorem{definition}[theorem]{Definition}
|
||||
@@ -76,6 +78,10 @@
|
||||
|
||||
\maketitle
|
||||
|
||||
\section*{Notation}
|
||||
|
||||
For a coloring $C : V(G) \to S$ and a color $c \in S$, we write $C^{-1}(c) = \{v \in V(G) : C(v) = c\}$ for the preimage of $c$ under $C$, i.e., the color class of $c$.
|
||||
|
||||
\section{Definitions}
|
||||
|
||||
\begin{definition}
|
||||
@@ -90,6 +96,16 @@ Equivalently, $L_i = \{v \in V(G) : d(v, u) = i\}$, where $d(v, u)$ denotes the
|
||||
Let $G$ be a maximal planar graph with a plane embedding, and let $\{L_0, L_1, L_2, \dots\}$ be the distance partition of $G$ from some $u \in V(G)$. The \emph{diamond scaffold} of $G$ relative to $u$ is the spanning subgraph $G^\diamond \subseteq G$ obtained by removing every edge $\{x, y\} \in E(G)$ such that $x, y \in L_i$ for some $i$.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}
|
||||
Let $G$ be a maximal planar graph. A \emph{plane diamond coloring} of $G$ is a proper $4$-coloring $C$ of $G$ such that there exist two colors $c_a, c_b$ and a diamond scaffold $G^\diamond$ of $G$ with a proper $2$-coloring $C^\diamond : V(G) \to \{c_a, c_b\}$ satisfying
|
||||
\[
|
||||
C^\diamond(v) = c_a \quad \text{for every } v \in C^{-1}(c_a),
|
||||
\]
|
||||
\[
|
||||
C^\diamond(v) = c_b \quad \text{for every } v \in C^{-1}(c_b).
|
||||
\]
|
||||
\end{definition}
|
||||
|
||||
\section{Results}
|
||||
|
||||
\begin{theorem}
|
||||
@@ -104,6 +120,47 @@ For any edge $\{x, y\} \in E(G)$, the depths of $x$ and $y$ differ by at most $1
|
||||
By construction, $G^\diamond$ contains no edge with both endpoints in the same level $L_i$. Combined with the bound above, every edge of $G^\diamond$ joins some $L_i$ to $L_{i+1}$. Color each vertex $v \in L_i$ by the parity of $i$. Every edge of $G^\diamond$ connects vertices of opposite parity, so this is a proper $2$-coloring.
|
||||
\end{proof}
|
||||
|
||||
\begin{lemma}
|
||||
The diamond scaffold $G^\diamond$ of a maximal planar graph $G$ relative to $u$ is connected.
|
||||
\end{lemma}
|
||||
|
||||
\begin{proof}
|
||||
Let $\{L_0, L_1, L_2, \dots\}$ be the distance partition of $G$ from $u$. We show by induction on $i$ that every vertex of $L_i$ is connected to $u$ in $G^\diamond$. The base case $i = 0$ is immediate, since $L_0 = \{u\}$. For $i \geq 1$, let $v \in L_i$. By definition of $L_i$, there is a shortest path from $v$ to $u$ of length $i$ in $G$, whose penultimate vertex $w$ lies in $L_{i-1}$. The edge $\{v, w\}$ joins $L_i$ to $L_{i-1}$, hence is not a level edge, hence belongs to $G^\diamond$. By the inductive hypothesis $w$ is connected to $u$ in $G^\diamond$, so $v$ is as well.
|
||||
\end{proof}
|
||||
|
||||
\begin{proposition}
|
||||
A maximal planar graph $G$ has a plane diamond coloring if and only if there exist a proper $4$-coloring $C$ of $G$, a vertex $u \in V(G)$, and two distinct colors $c_a, c_b$ such that, with respect to the distance partition $\{L_0, L_1, L_2, \dots\}$ of $G$ from $u$,
|
||||
\[
|
||||
C^{-1}(c_a) \subseteq \bigcup_{i \text{ even}} L_i \qquad \text{and} \qquad C^{-1}(c_b) \subseteq \bigcup_{i \text{ odd}} L_i.
|
||||
\]
|
||||
\end{proposition}
|
||||
|
||||
\begin{proof}
|
||||
Since $G^\diamond$ is connected and bipartite (Theorem 2.1 and Lemma 2.2), its proper $2$-coloring is unique up to swapping the two colors, and is given by the parity of level. Hence a proper $2$-coloring $C^\diamond : V(G) \to \{c_a, c_b\}$ of $G^\diamond$ exists with $C^\diamond(v) = c_a$ on the even-parity layers and $C^\diamond(v) = c_b$ on the odd-parity layers (or vice versa). The agreement condition $C(v) = C^\diamond(v)$ on $C^{-1}(c_a) \cup C^{-1}(c_b)$ is then equivalent to the stated containment.
|
||||
\end{proof}
|
||||
|
||||
\begin{remark}
|
||||
The conjecture below asserts a structural property of $4$-colorings of maximal planar graphs strictly stronger than the conclusion of the Four Color Theorem \cite{appel1977every,robertson1997four}: it requires not merely the existence of a proper $4$-coloring, but the existence of a proper $4$-coloring together with a root $u$ such that two of the four color classes are separated by the parity of the BFS layering from $u$.
|
||||
\end{remark}
|
||||
|
||||
\begin{conjecture}
|
||||
Every maximal planar graph $G$ has a plane diamond coloring.
|
||||
\end{conjecture}
|
||||
|
||||
\begin{thebibliography}{9}
|
||||
|
||||
\bibitem{appel1977every}
|
||||
K.~Appel and W.~Haken,
|
||||
\emph{Every planar map is four colorable},
|
||||
Illinois Journal of Mathematics, vol.~21, no.~3, pp.~429--567, 1977.
|
||||
|
||||
\bibitem{robertson1997four}
|
||||
N.~Robertson, D.~Sanders, P.~Seymour, and R.~Thomas,
|
||||
\emph{The four-colour theorem},
|
||||
Journal of Combinatorial Theory, Series B, vol.~70, no.~1, pp.~2--44, 1997.
|
||||
|
||||
\end{thebibliography}
|
||||
|
||||
\end{document}
|
||||
|
||||
%-----------------------------------------------------------------------
|
||||
|
||||
Reference in New Issue
Block a user