極大平面圖的結(jié)構(gòu)與著色理論 (1)色多項式遞推公式與四色猜想

許進

doi:10.11999/JEIT160072

極大平面圖的結(jié)構(gòu)與著色理論 (1)色多項式遞推公式與四色猜想

doi: 10.11999/JEIT160072

許進¹

基金項目:

國家973規(guī)劃項目(2013CB329600)，國家自然科學(xué)基金(61472012, 6152046, 6152012, 61572492, 61372191, 61472012)

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- 文章訪問數(shù): 2660
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- 被引次數(shù): 0
出版歷程
- 收稿日期: 2016-01-15
- 修回日期: 2016-01-18
- 刊出日期: 2016-04-19

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

XU Jin¹

Funds:

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

摘要

摘要: 該文給出了極大平面圖$G$的色多項式遞推計算公式：若$\delta(G)=4$, $W_4^\nu$是$G$中輪心為$\nu$，輪圈為$\nu_1\nu_2\nu_3\nu_4\nu_1$的4-輪，則$f(G,4)=f(G_1,4)+f(G_2,4)$，其中$G_1=(G-\nu)\circ{\nu_1,\nu_3}$, $G_2=(G-\nu)\circ{\nu_2,\nu_4}$；若$\delta(G)=5$,$W_5^\nu$是$G$中$\nu$為輪心，以$\nu_1\nu_2\nu_3\nu_4\nu_5\nu_1$為輪圈的5-輪，則$f(G,4)=[f(G_1,4)-f(G_1\cup{\nu_1\nu_4,\nu_1\nu_3},4)] +[f(G_2,4)-f(G_2\cup {\nu_3\nu_1,\nu_3\nu_5},4)]+ [f(G_3,4)-f(G_3\cup {\nu_1\nu_4},4)]$，其中$G_1=(G-\nu)\circ{\nu_2,\nu_5}$, $G_2=(G-\nu)\circ{\nu_2,\nu_4}$, $G_3=(G-\nu)\circ{\nu_3,\nu_5}$，“$\circ$”表示收縮運算；進而討論了使用公式證明四色猜想的應(yīng)用：將四色猜想轉(zhuǎn)化成研究一種特殊圖類：4-色漏斗型偽唯一4-色極大平面圖。
- 四色猜想 /
- 極大平面圖 /
- 色多項式 /
- 偽唯一4-色平面圖 /
- 4-色漏斗
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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