FourColorTheorem2 2009-02-21 11:10:55 2009-02-21 12:32:07 Theorem chromatic number of a graph the four-color theorem chromatic number of a graph % this is the default PlanetPhysics preamble. as your % almost certainly you want these \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym, enumerate} \usepackage{xypic, xspace} \usepackage[mathscr]{eucal} \usepackage[dvips]{graphicx} \usepackage[curve]{xy} % define commands here \theoremstyle{plain} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{example}{Example}[section] %\theoremstyle{remark} \newtheorem{remark}{Remark}[section] \newtheorem*{notation}{Notation} \newtheorem*{claim}{Claim} \renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote}}} \numberwithin{equation}{section} \newcommand{\Ad}{{\rm Ad}} \newcommand{\Aut}{{\rm Aut}} \newcommand{\Cl}{{\rm Cl}} \newcommand{\Co}{{\rm Co}} \newcommand{\DES}{{\rm DES}} \newcommand{\Diff}{{\rm Diff}} \newcommand{\Dom}{{\rm Dom}} \newcommand{\Hol}{{\rm Hol}} \newcommand{\Mon}{{\rm Mon}} \newcommand{\Hom}{{\rm Hom}} \newcommand{\Ker}{{\rm Ker}} \newcommand{\Ind}{{\rm Ind}} \newcommand{\IM}{{\rm Im}} \newcommand{\Is}{{\rm Is}} \newcommand{\ID}{{\rm id}} \newcommand{\grpL}{{\rm GL}} \newcommand{\Iso}{{\rm Iso}} \newcommand{\rO}{{\rm O}} \newcommand{\Sem}{{\rm Sem}} \newcommand{\SL}{{\rm Sl}} \newcommand{\St}{{\rm St}} \newcommand{\Sym}{{\rm Sym}} \newcommand{\Symb}{{\rm Symb}} \newcommand{\SU}{{\rm SU}} \newcommand{\Tor}{{\rm Tor}} \newcommand{\U}{{\rm U}} \newcommand{\A}{\mathcal A} \newcommand{\Ce}{\mathcal C} \newcommand{\D}{\mathcal D} \newcommand{\E}{\mathcal E} \newcommand{\F}{\mathcal F} %\newcommand{\grp}{\mathcal G} \renewcommand{\H}{\mathcal H} \renewcommand{\cL}{\mathcal L} \newcommand{\Q}{\mathcal Q} \newcommand{\R}{\mathcal R} \newcommand{\cS}{\mathcal S} \newcommand{\cU}{\mathcal U} \newcommand{\W}{\mathcal W} \newcommand{\bA}{\mathbb{A}} \newcommand{\bB}{\mathbb{B}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bD}{\mathbb{D}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bG}{\mathbb{G}} \newcommand{\bK}{\mathbb{K}} \newcommand{\bM}{\mathbb{M}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bO}{\mathbb{O}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bV}{\mathbb{V}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bfE}{\mathbf{E}} \newcommand{\bfX}{\mathbf{X}} \newcommand{\bfY}{\mathbf{Y}} \newcommand{\bfZ}{\mathbf{Z}} \renewcommand{\O}{\Omega} \renewcommand{\o}{\omega} \newcommand{\vp}{\varphi} \newcommand{\vep}{\varepsilon} \newcommand{\diag}{{\rm diag}} \newcommand{\grp}{{\mathsf{G}}} \newcommand{\dgrp}{{\mathsf{D}}} \newcommand{\desp}{{\mathsf{D}^{\rm{es}}}} \newcommand{\grpeod}{{\rm Geod}} %\newcommand{\grpeod}{{\rm geod}} \newcommand{\hgr}{{\mathsf{H}}} \newcommand{\mgr}{{\mathsf{M}}} \newcommand{\ob}{{\rm Ob}} \newcommand{\obg}{{\rm Ob(\mathsf{G)}}} \newcommand{\obgp}{{\rm Ob(\mathsf{G}')}} \newcommand{\obh}{{\rm Ob(\mathsf{H})}} \newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}} \newcommand{\grphomotop}{{\rho_2^{\square}}} \newcommand{\grpcalp}{{\mathsf{G}(\mathcal P)}} \newcommand{\rf}{{R_{\mathcal F}}} \newcommand{\grplob}{{\rm glob}} \newcommand{\loc}{{\rm loc}} \newcommand{\TOP}{{\rm TOP}} \newcommand{\wti}{\widetilde} \newcommand{\what}{\widehat} \renewcommand{\a}{\alpha} \newcommand{\be}{\beta} \newcommand{\grpa}{\grpamma} %\newcommand{\grpa}{\grpamma} \newcommand{\de}{\delta} \newcommand{\del}{\partial} \newcommand{\ka}{\kappa} \newcommand{\si}{\sigma} \newcommand{\ta}{\tau} \newcommand{\lra}{{\longrightarrow}} \newcommand{\ra}{{\rightarrow}} \newcommand{\rat}{{\rightarrowtail}} \newcommand{\ovset}[1]{\overset {#1}{\ra}} \newcommand{\ovsetl}[1]{\overset {#1}{\lra}} \newcommand{\hr}{{\hookrightarrow}} \newcommand{\<}{{\langle}} \def\baselinestretch{1.1} \hyphenation{prod-ucts} \newcommand{\sqdiagram}[9]{$$ \diagram #1 \rto^{#2} \dto_{#4}& #3 \dto^{#5} \\ #6 \rto_{#7} & #8 \enddiagram \eqno{\mbox{#9}}$$ } \def\C{C^{\ast}} \newcommand{\labto}[1]{\stackrel{#1}{\longrightarrow}} \newcommand{\quadr}[4] {\begin{pmatrix} & #1& \\[-1.1ex] #2 & & #3\\[-1.1ex]& #4& \end{pmatrix}} \def\D{\mathsf{D}} \section{The four-color theorem} \begin{theorem} Any map in a plane can be colored using four-colors in such a way that regions sharing a common boundary --that is not a single point-- do not share the same color. \end{theorem} \subsection{History} F. Guthrie, first conjectured the four color theorem in 1853. The first published paper on Guthrie's conjecture was not however published until 1878 by Cayley. Appel and Haken published 1977 from the University of Illinois at Urbana- Champaign a computer-assisted proof that four colors are sufficient. However, some mathematicians do not accept it because it utilized examination of cases assisted by a computer, with the posssibility of software errors always remaining. On the other hand, no flaws have yet been found, in spite of repeated attempts. The first independent proof of the four color theorem was constructed by Robertson et al. in 1996 and by Thomas in 1998. Then, in December 2004, G. Gonthier in Cambridge, England --colaborating with B. Werner of INRIA in France -- announced that they were able to validate the Robertson et al. (1996) proof of the color theorem by formulating it in the equational logic program called ``Coq'' (gaelic ?), and then were able to confirm the validity of each of its steps as reported by Devlin in 2005 and Knight in 2005. \subsection{Extensions} The Heawood conjecture is a more general proposition for map coloring, stating that in a genus $0$ space, including both the sphere or plane, four colors would suffice. On the other hand, for any genus $> 0$, Ringel and Youngs were able to prove in a report published in 1968 the following theorem: \begin{theorem} The Heawood conjecture specifies the correct necessary number of colors for any genus $0$, except for the Klein bottle. However, the correct number of colors for any Klein bottle is six --not seven colors as stated by Heawood. Thus, in general for any genus $0$ the coloring number is no greater than six. Furthermore, The chromatic number of a surface of genus $g$ is given by the formula $$\gamma(g) = [1/2 (7 + \sqrt{48g +1})],$$ where the righ-hand-side is called the \emph{floor function}. \end{theorem} A closely related theme is that of graph coloring using computer algorithms. \subsubsection{Graph Coloring utilizing Computer Algorithms} The \emph{chromatic number of a graph} is the smallest number of colors needed to color the vertices of so that no two adjacent vertices share the same color (p. 210 in Skiena 1990); this is the smallest value of possible to obtain a $k$--coloring. The chromatic number of a graph $\tilde{G}$ can be computed as the smallest positive integer $z$ such that the chromatic polynomial $\pi_{\tilde{G}}(z) >0 $; thus, calculating the chromatic number of a graph is an NP-complete problem (Skiena 1990, pp. 211-212), but no general algorithm has been found yet for any arbitrary graph as suggested by Harary (1994, p. 127). Erd\"os (1959) proved that there are graphs with arbitrarily large girth and chromatic numbers (cited in Bollob\'as and West, 2000). Chromatic numbers and minimal colorings for many colored graphs are readily illustrated by employing Mathematica$^{TM}$ as shown at the \PMlinkexternal{mathworld website.}{http://mathworld.wolfram.com/ChromaticNumber.html} As an example, the chromatic number can be digitally computed using ChromaticNumber$[g]$ in the Mathematica$^{TM}$ package ``Combinatorica''; minimal coloring can also be computed by using MinimalColoring$[g]$ in the same package. Pre-computed chromatic numbers are readily available for most remarkable or special-property graphs can be obtained using $GraphData$, (with $[graph,\, ``ChromaticNumber'']$). \begin{thebibliography}{99} \bibitem{BW2k} Bollob\'as, B. and West, D. B. ``A Note on Generalized Chromatic Number and Generalized Girth.'' {\em Discr. Math.}, 213, 29-34, 2000. \bibitem{ED2k} Eppstein, D. \PMlinkexternal{The Chromatic Number of the Plane.}{http://www.ics.uci.edu/~eppstein/junkyard/plane-color.html}. \bibitem{EP59} Erd\"os, ``P. Graph Theory and Probability.'' Canad. J. Math. 11, 34-38, 1959. \bibitem{EP61} Erd\"os, P. ``Graph Theory and Probability II.'' Canad. J. Math. 13, 346-352, 1961. \bibitem{HF94} Harary, F. {\em Graph Theory}. Reading, MA: Addison--Wesley, 1994. \bibitem{LL68} Lov\'asz, L. ``On (the) Chromatic Number of Finite Set-Systems.'', Acta Math. Acad. Sci. Hungar. 19, 59-67, 1968. \bibitem{HF94} Skiena, S. {\em Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica}. Reading, MA: Addison--Wesley, 1990. Sloane, N. J. A. Sequences $A000012/M0003, A000934/M3292, A068917, A068918, and A068919$ in {\em The On-Line Encyclopedia of Integer Sequences.} Weisstein, Eric W. ``Chromatic Number.'', In \PMlinkexternal{\emph{MathWorld--A Wolfram Web Resource}.}{http://mathworld.wolfram.com/ChromaticNumber.html} \end{thebibliography}