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, 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.
Citation
Nicolas Roussel. "Local Condition for Planar Graphs of Maximum Degree 6 to be Total 8-Colorable." Taiwanese J. Math. 15 (1) 87 - 99, 2011. https://doi.org/10.11650/twjm/1500406163
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