Theory on the Structure and Coloring of Maximal Planar Graphs (1)Recursion Formula of Chromatic Polynomial and FourColor Conjecture

XU Jin

doi:10.11999/JEIT160072

Volume 38 Issue 4

Apr. 2016

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XU Jin. Theory on the Structure and Coloring of Maximal Planar Graphs (1)Recursion Formula of Chromatic Polynomial and FourColor Conjecture[J]. Journal of Electronics & Information Technology, 2016, 38(4): 763-779. doi: 10.11999/JEIT160072

Citation:

XU Jin. Theory on the Structure and Coloring of Maximal Planar Graphs (1)Recursion Formula of Chromatic Polynomial and FourColor Conjecture[J]. Journal of Electronics & Information Technology, 2016, 38(4): 763-779. doi: 10.11999/JEIT160072

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Theory on the Structure and Coloring of Maximal Planar Graphs (1)Recursion Formula of Chromatic Polynomial and FourColor Conjecture

doi: 10.11999/JEIT160072

XU Jin¹

Funds:

The National 973 Program of China (2013CB329600), The National Natural Science Foundation of China (61472012, 6152046, 6152012, 61572492, 61372191, 61472012)

Received Date: 2016-01-15
Rev Recd Date: 2016-01-18
Publish Date: 2016-04-19

Abstract

Abstract

In this paper, two recursion formulae of chromatic polynomial of a maximal planar graph $G$ are obtained: when $\delta(G)=4$, let $W_4^\nu$ be a 4-wheel of $G$ with wheel-center $\nu$ and wheel-cycle $\nu_1\nu_2\nu_3\nu_4\nu_1$, then $f(G,4)=f((G,4)\circ{\nu_1,\nu_3},4)+ f((G,4)\circ{\nu_2,\nu_4},4)$; when $\delta(G)=5$, let $W_5^\nu$ a 5-wheel of $G$ with wheel-center $\nu$ and wheel-cycle $\nu_1\nu_2\nu_3\nu_4\nu_5\nu_1$, then $f(G,4)=[f(G_1,4)-f(G_1\cup{\nu_1\nu_4,\nu_1\nu_3},4)] +[f(G_2,4)-f(G_2\cup {\nu_3\nu_1,\nu_3\nu_5},4)]+ [f(G_3,4)-f(G_3\cup {\nu_1\nu_4},4)]$, $G_1=(G-\nu)\circ{\nu_2,\nu_5}$, $G_2=(G-\nu)\circ{\nu_2,\nu_4}$, $G_3=(G-\nu)\circ{\nu_3,\nu_5}$, where $“\circ”$ denotes the operation of vertex contraction. Moreover, the application of the above formulae to the proof of Four-Color Conjecture is investigated. By using these formulae, the proof of Four-Color Conjecture boils down to the study on a special class of graphs, viz., 4-chromatic-funnel pseudo uniquely-4-colorable maximal planar graphs.
- Four-Color Conjecture,
- Maximal planar graphs,
- Chromatic polynomial,
- Pseudo uniquely-4-colorable planar graphs,
- 4-chromatic-funnel

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References(35)

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