Local Condition for Planar Graphs of Maximum Degree 6 to be Total 8-Colorable

Abstract

Recently Sun et al [X.-Y. Sun, J.-L. Wu, Y.-W. Wu, J.-F. Hou, Total colorings of planar graphs without adjacent triangles, {\em Discrete Math 309:202-206} (2009)] proved that planar graphs with maximum degree six and with no adjacent triangles are total $8$-colorable. This results implies that if every vertex of a planar graph of maximum degree six is missing either a $3$-cycle or a $4$-cycle, then the graph is total $8$-colorable. In this paper we strengthen that condition by showing that if every vertex of a planar graph of maximum degree six is missing some $k_v$-cycle for $k_v\in\{3,4,5,6,7,8\}$, then the graph is total $8$-colorable.