Plane graphs with maximum degree 7 and without 5-cycles with chords are 8-totally-colorable
Qiang SUN, Xin Tao, 岚 沈, YingQian WANG · Scientia Sinica Mathematica · 2011
Let G = (V,E) be a graph with the set of vertices V and the set of edges E. If one can use k colors to color the elements in V ∪ E such that any pair of adjacent or incident elements receive distinct colors, then G is said to be k-totally-colorable. Clearly, at least Δ + 1 colors are needed to color a graph totally, where Δ is the maximum degree of G. It is known that the plane graphs with maximum degree Δ > 8 and without 5-cycles with chords are (Δ + 1)-totally-colorable. In this paper, we prove that the plane graphs with maximum degree 7 and without 5-cycles with chords are 8-totally-colorable.