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Erschienen in:
03.09.2016
Total coloring of planar graphs without adjacent chordal 6-cycles
Erschienen in: Journal of Combinatorial Optimization | Ausgabe 1/2017
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Abstract
A total coloring of a graph G is a coloring such that no two adjacent or incident elements receive the same color. In this field there is a famous conjecture, named Total Coloring Conjecture, saying that the the total chromatic number of each graph G is at most \(\Delta +2\). Let G be a planar graph with maximum degree \(\Delta \ge 7\) and without adjacent chordal 6-cycles, that is, two cycles of length 6 with chord do not share common edges. In this paper, it is proved that the total chromatic number of G is \(\Delta +1\), which partly confirmed Total Coloring Conjecture.