Add Motivation section and restore diamond scaffold definition
Frames the paper around the scaffold-first 4-coloring program: 2-color
the BFS-layered bipartite spanning subgraph (the diamond scaffold),
then promote select vertices with two new colors to absorb the
discarded same-layer edges. Reintroduces the diamond scaffold
definition (removed in b5a9030 along with the equivalence machinery)
since it now plays a motivational rather than definitional role.
Replaces hardcoded definition/theorem/conjecture numbers with stable
\ref{}-based cross-references.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
@@ -1,19 +1,27 @@
|
||||
\relax
|
||||
\citation{appel1977every}
|
||||
\citation{robertson1997four}
|
||||
\citation{mckaygraph6}
|
||||
\@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 }
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The unique smallest maximal planar graph with no plane diamond coloring; it has $13$ vertices and degree sequence $(6,6,6,6,6,6,6,5,5,4,4,3,3)$.}}{2}{}\protected@file@percent }
|
||||
\newlabel{fig:counterexample}{{1}{2}}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Motivation}}{1}{}\protected@file@percent }
|
||||
\citation{appel1977every}
|
||||
\citation{robertson1997four}
|
||||
\citation{mckaygraph6}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Definitions}}{2}{}\protected@file@percent }
|
||||
\newlabel{def:distance-partition}{{2.1}{2}}
|
||||
\newlabel{def:scaffold}{{2.2}{2}}
|
||||
\newlabel{def:diamond}{{2.3}{2}}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Results}}{2}{}\protected@file@percent }
|
||||
\newlabel{thm:counterexample}{{3.3}{2}}
|
||||
\bibcite{appel1977every}{1}
|
||||
\bibcite{robertson1997four}{2}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The unique smallest maximal planar graph with no plane diamond coloring; it has $13$ vertices and degree sequence $(6,6,6,6,6,6,6,5,5,4,4,3,3)$.}}{3}{}\protected@file@percent }
|
||||
\newlabel{fig:counterexample}{{1}{3}}
|
||||
\newlabel{conj:mindeg5}{{3.4}{3}}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{3}{}\protected@file@percent }
|
||||
\bibcite{mckaygraph6}{3}
|
||||
\newlabel{tocindent-1}{0pt}
|
||||
\newlabel{tocindent0}{12.7778pt}
|
||||
\newlabel{tocindent1}{17.77782pt}
|
||||
\newlabel{tocindent2}{0pt}
|
||||
\newlabel{tocindent3}{0pt}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{3}{}\protected@file@percent }
|
||||
\gdef \@abspage@last{3}
|
||||
\gdef \@abspage@last{4}
|
||||
|
||||
@@ -1,12 +1,12 @@
|
||||
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 9 MAY 2026 13:18
|
||||
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 9 MAY 2026 13:44
|
||||
entering extended mode
|
||||
restricted \write18 enabled.
|
||||
file:line:error style messages enabled.
|
||||
%&-line parsing enabled.
|
||||
**/Users/didericis/Code/math-research/papers/plane_diamond_coloring/paper.tex
|
||||
(/Users/didericis/Code/math-research/papers/plane_diamond_coloring/paper.tex
|
||||
**paper.tex
|
||||
(./paper.tex
|
||||
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
|
||||
\linespacing=\dimen138
|
||||
\normalparindent=\dimen139
|
||||
@@ -18,14 +18,17 @@ Package: amsmath 2021/10/15 v2.17l AMS math features
|
||||
For additional information on amsmath, use the `?' option.
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
|
||||
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
|
||||
\@emptytoks=\toks16
|
||||
\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
|
||||
\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
|
||||
)
|
||||
\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 2939.
|
||||
)
|
||||
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
|
||||
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
|
||||
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
|
||||
\symAMSa=\mathgroup4
|
||||
\symAMSb=\mathgroup5
|
||||
@@ -100,34 +106,43 @@ LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold'
|
||||
\thm@postskip=\skip55
|
||||
\thm@headsep=\skip56
|
||||
\dth@everypar=\toks26
|
||||
) (/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)
|
||||
(/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)
|
||||
\KV@toks@=\toks27
|
||||
) (/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)
|
||||
(/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)
|
||||
) (/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
|
||||
)
|
||||
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
|
||||
))
|
||||
\Gin@req@height=\dimen150
|
||||
\Gin@req@width=\dimen151
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/latex/url/url.sty
|
||||
)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/url/url.sty
|
||||
\Urlmuskip=\muskip17
|
||||
Package: url 2013/09/16 ver 3.4 Verb mode for urls, etc.
|
||||
)
|
||||
\c@theorem=\count272
|
||||
(/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)
|
||||
\l__color_backend_stack_int=\count273
|
||||
\l__pdf_internal_box=\box53
|
||||
) (./paper.aux)
|
||||
)
|
||||
(./paper.aux)
|
||||
\openout1 = `paper.aux'.
|
||||
|
||||
LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 57.
|
||||
@@ -145,13 +160,17 @@ LaTeX Font Info: ... okay on input line 57.
|
||||
LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 57.
|
||||
LaTeX Font Info: ... okay on input line 57.
|
||||
LaTeX Font Info: Trying to load font information for U+msa on input line 57.
|
||||
|
||||
(/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 57.
|
||||
(/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
|
||||
) (/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).]
|
||||
\scratchcounter=\count274
|
||||
\scratchdimen=\dimen152
|
||||
@@ -166,29 +185,49 @@ File: umsb.fd 2013/01/14 v3.01 AMS symbols B
|
||||
\everyMPtoPDFconversion=\toks29
|
||||
) (/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 Info: Redefining graphics rule for `.eps' on input line 485.
|
||||
(/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
|
||||
)) [1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}]
|
||||
<counterexample.png, id=19, 338.9463pt x 339.669pt>
|
||||
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
|
||||
File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv
|
||||
e
|
||||
))
|
||||
[1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}]
|
||||
[2]
|
||||
<counterexample.png, id=22, 338.9463pt x 339.669pt>
|
||||
File: counterexample.png Graphic file (type png)
|
||||
<use counterexample.png>
|
||||
Package pdftex.def Info: counterexample.png used on input line 144.
|
||||
Package pdftex.def Info: counterexample.png used on input line 158.
|
||||
(pdftex.def) Requested size: 161.9989pt x 162.34474pt.
|
||||
[2 <./counterexample.png>] [3] (./paper.aux) )
|
||||
[3 <./counterexample.png>] [4] (./paper.aux) )
|
||||
Here is how much of TeX's memory you used:
|
||||
2670 strings out of 478268
|
||||
38814 string characters out of 5846347
|
||||
339139 words of memory out of 5000000
|
||||
20714 multiletter control sequences out of 15000+600000
|
||||
2674 strings out of 478268
|
||||
38703 string characters out of 5846347
|
||||
343189 words of memory out of 5000000
|
||||
20719 multiletter control sequences out of 15000+600000
|
||||
476338 words of font info for 57 fonts, out of 8000000 for 9000
|
||||
1302 hyphenation exceptions out of 8191
|
||||
69i,8n,76p,836b,344s 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/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/cm/cmtt10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb>
|
||||
Output written on paper.pdf (3 pages, 216857 bytes).
|
||||
69i,8n,76p,867b,344s stack positions out of 10000i,1000n,20000p,200000b,200000s
|
||||
</usr/local/texlive/2022/texmf-d
|
||||
ist/fonts/type1/public/amsfonts/cm/cmbx10.pfb></usr/local/texlive/2022/texmf-di
|
||||
st/fonts/type1/public/amsfonts/cm/cmcsc10.pfb></usr/local/texlive/2022/texmf-di
|
||||
st/fonts/type1/public/amsfonts/cm/cmex10.pfb></usr/local/texlive/2022/texmf-dis
|
||||
t/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/f
|
||||
onts/type1/public/amsfonts/cm/cmr10.pfb></usr/local/texlive/2022/texmf-dist/fon
|
||||
ts/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/typ
|
||||
e1/public/amsfonts/cm/cmsy10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type
|
||||
1/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/p
|
||||
ublic/amsfonts/cm/cmti8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/pub
|
||||
lic/amsfonts/cm/cmtt10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/publ
|
||||
ic/amsfonts/cm/cmtt8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public
|
||||
/amsfonts/symbols/msam10.pfb>
|
||||
Output written on paper.pdf (4 pages, 223251 bytes).
|
||||
PDF statistics:
|
||||
92 PDF objects out of 1000 (max. 8388607)
|
||||
54 compressed objects within 1 object stream
|
||||
95 PDF objects out of 1000 (max. 8388607)
|
||||
56 compressed objects within 1 object stream
|
||||
0 named destinations out of 1000 (max. 500000)
|
||||
6 words of extra memory for PDF output out of 10000 (max. 10000000)
|
||||
|
||||
|
||||
Binary file not shown.
@@ -85,9 +85,19 @@
|
||||
|
||||
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{Motivation}
|
||||
|
||||
Let $G$ be a maximal planar graph. By the Four Color Theorem \cite{appel1977every,robertson1997four}, $G$ admits a proper $4$-coloring; the question that motivates this paper is whether such a coloring can always be exhibited via a particular structural construction, rather than as the output of an ad hoc case analysis.
|
||||
|
||||
The construction we have in mind proceeds as follows. Fix a root vertex $u \in V(G)$ and consider the BFS layering $\{L_0, L_1, L_2, \dots\}$ of $G$ from $u$ (Definition~\ref{def:distance-partition}). Remove from $G$ every edge whose two endpoints lie in the same layer $L_i$, producing a spanning subgraph $G^\diamond \subseteq G$ which we call the \emph{diamond scaffold} of $G$ relative to $u$ (Definition~\ref{def:scaffold}). For any edge $\{x, y\} \in E(G)$ the BFS depths of $x$ and $y$ differ by at most $1$, so every edge surviving in $G^\diamond$ joins some level $L_i$ to $L_{i+1}$; in particular $G^\diamond$ is bipartite, and the parity of the layer index supplies a canonical proper $2$-coloring of $G^\diamond$ using two colors, which we denote $c_a$ and $c_b$.
|
||||
|
||||
To extend this $2$-coloring of the scaffold to a proper $4$-coloring of the original graph $G$, we must dispose of the edges discarded in passing from $G$ to $G^\diamond$ --- namely the edges whose endpoints share a BFS layer. The natural strategy is to recolor a chosen subset of vertices using two new colors $c_c, c_d$, just enough to resolve the discarded same-layer conflicts while preserving the canonical $\{c_a, c_b\}$-coloring on the remaining vertices. A \emph{plane diamond coloring} of $G$ (Definition~\ref{def:diamond}) is precisely a proper $4$-coloring of $G$ obtained in this way: two of its color classes, $C^{-1}(c_a)$ and $C^{-1}(c_b)$, are confined to even-indexed and odd-indexed BFS layers respectively, so that they extend the canonical $2$-coloring of the diamond scaffold $G^\diamond$ relative to some root $u$.
|
||||
|
||||
This paper investigates which maximal planar graphs admit such a coloring. The Four Color Theorem guarantees a proper $4$-coloring; the diamond coloring asks for one obeying the additional structural constraint above. Theorem~\ref{thm:counterexample} shows the constraint genuinely fails on some triangulations, with a unique smallest obstruction of order $13$; Conjecture~\ref{conj:mindeg5} asserts that the obstructions disappear once $\delta(G) \geq 5$.
|
||||
|
||||
\section{Definitions}
|
||||
|
||||
\begin{definition}
|
||||
\begin{definition} \label{def:distance-partition}
|
||||
Let $G$ be a graph and let $u \in V(G)$. The \emph{distance partition} of $G$ from $u$ is the partition $\{L_0, L_1, L_2, \dots\}$ of $V(G)$ obtained by breadth-first search from $u$:
|
||||
\[
|
||||
L_0 = \{u\}, \qquad L_{i+1} = \{v \in V(G) \setminus (L_0 \cup \cdots \cup L_i) : v \text{ is adjacent to some } w \in L_i\}.
|
||||
@@ -95,7 +105,11 @@ Let $G$ be a graph and let $u \in V(G)$. The \emph{distance partition} of $G$ fr
|
||||
Equivalently, $L_i = \{v \in V(G) : d(v, u) = i\}$, where $d(v, u)$ denotes the graph distance between $v$ and $u$ in $G$. We call each $L_i$ the \emph{$i$-th level} of the partition.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}
|
||||
\begin{definition} \label{def:scaffold}
|
||||
Let $G$ be a maximal planar graph and let $u \in V(G)$, with distance partition $\{L_0, L_1, L_2, \dots\}$ from $u$. The \emph{diamond scaffold} of $G$ relative to $u$ is the spanning subgraph $G^\diamond \subseteq G$ obtained from $G$ by removing every edge $\{x, y\} \in E(G)$ such that $x, y \in L_i$ for some $i$.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition} \label{def:diamond}
|
||||
Let $G$ be a maximal planar graph. A \emph{plane diamond coloring} of $G$ is a proper $4$-coloring $C$ of $G$ for which there exist 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.
|
||||
@@ -105,14 +119,14 @@ Let $G$ be a maximal planar graph. A \emph{plane diamond coloring} of $G$ is a p
|
||||
\section{Results}
|
||||
|
||||
\begin{remark}
|
||||
Definition 1.2 imposes a structural condition on $4$-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 $u$.
|
||||
Definition~\ref{def:diamond} imposes a structural condition on $4$-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{theorem}
|
||||
\begin{theorem} \label{thm:counterexample}
|
||||
The preceding conjecture is false. Moreover, the smallest counterexample has order $13$, and is unique up to isomorphism among triangulations of order at most $13$.
|
||||
\end{theorem}
|
||||
|
||||
@@ -130,7 +144,7 @@ shown in Figure~\ref{fig:counterexample}. Equivalently, $G$ has edge set
|
||||
\end{align*}
|
||||
We have $|V(G)| = 13$ and $|E(G)| = 33 = 3 \cdot 13 - 6$, so $G$ is a triangulation.
|
||||
|
||||
By Definition 1.2, it suffices to show that for every root $u \in V(G)$, no proper $4$-coloring $C$ of $G$ admits two distinct colors $c_a, c_b$ with $C^{-1}(c_a)$ contained in the union of even-indexed levels and $C^{-1}(c_b)$ contained in the union of odd-indexed levels of the distance partition from $u$.
|
||||
By Definition~\ref{def:diamond}, it suffices to show that for every root $u \in V(G)$, no proper $4$-coloring $C$ of $G$ admits two distinct colors $c_a, c_b$ with $C^{-1}(c_a)$ contained in the union of even-indexed levels and $C^{-1}(c_b)$ contained in the union of odd-indexed levels of the distance partition from $u$.
|
||||
|
||||
For a fixed root $u$, the existence of such a triple $(C, c_a, c_b)$ is equivalent to $4$-colorability of the auxiliary graph $H_u$ obtained from $G$ by adjoining two new vertices $\alpha, \beta$, joining $\alpha$ to every vertex in odd-indexed levels, joining $\beta$ to every vertex in even-indexed levels, and adding the edge $\{\alpha, \beta\}$. Indeed, in any proper $4$-coloring of $H_u$ the colors of $\alpha$ and $\beta$ are distinct and absent from the odd-parity and even-parity layers of $G$ respectively, yielding $c_a := C(\alpha)$ and $c_b := C(\beta)$. Conversely, given a $4$-coloring satisfying the parity-separation condition, setting $C(\alpha) := c_a$ and $C(\beta) := c_b$ extends it to a proper $4$-coloring of $H_u$.
|
||||
|
||||
@@ -146,12 +160,12 @@ For minimality and uniqueness, we exhaustively enumerated every maximal planar g
|
||||
\label{fig:counterexample}
|
||||
\end{figure}
|
||||
|
||||
\begin{conjecture}
|
||||
\begin{conjecture} \label{conj:mindeg5}
|
||||
Every maximal planar graph $G$ of minimum degree at least $5$ has a plane diamond coloring.
|
||||
\end{conjecture}
|
||||
|
||||
\begin{remark}
|
||||
We have verified Conjecture 2.4 computationally for all maximal planar graphs of minimum degree at least $5$ and order at most $N$, by exhaustive enumeration via \texttt{Sage}'s \texttt{graphs.planar\_graphs} generator and the auxiliary-graph reduction described in the proof of Theorem 2.3. No counterexample has been found.
|
||||
We have verified Conjecture~\ref{conj:mindeg5} computationally for all maximal planar graphs of minimum degree at least $5$ and order at most $N$, by exhaustive enumeration via \texttt{Sage}'s \texttt{graphs.planar\_graphs} generator and the auxiliary-graph reduction described in the proof of Theorem~\ref{thm:counterexample}. No counterexample has been found.
|
||||
\end{remark}
|
||||
|
||||
\begin{thebibliography}{9}
|
||||
|
||||
Reference in New Issue
Block a user