关于最大度为7的平面图全染色的一个注记

A note on total coloring of plane graphs with maximum degree seven

给最大度为Δ的图进行全染色至少要用Δ+1种颜色.全染色猜想断言每个图都是(Δ+2)-全可染的.但即使对于平面图,全染色猜想依然未得到证实.在该研究方向已证明满足下述条件之一的最大度为Δ的平面图是(Δ+1)-全可染的:1)Δ≥9;2)Δ=8且不含相邻三角形.证明了最大度为7且不含带弦4-圈和带弦5-圈的平面图是8-全可染的.该结果进一步拓展了(Δ+1)-全可染平面图类.

To color a graph with maximum degree Δ totally,at least Δ＋1 colors are needed.The total coloring conjecture asserts that every graph is（Δ＋2）-totally-colorable.Even for plane graphs,this conjecture remains open.It had been proved so far that plane graphs with maximum degree Δ could be（Δ＋1）-totally-colorable if they satisfied one of the following conditions： 1） Δ≥9;2） Δ=8 and had no adjacent triangles.It was proved that a plane graph with maximum degree 7 could be 8-totally-colorable if it had neither chordal 4-cycle nor chordal 5-cycle.This result extended the known class of（Δ＋1）-totally-colorable plane graphs.