Planar graphs with $\Delta =9$ are neighbor-distinguishing totally 12-colorable

Abstract

The neighbor-distinguishing total coloring of a graph G is a proper total coloring of G using k colors such that any two adjacent vertices have different sets of colors. It was known that every planar graph G with $\Delta \ge 10$ is neighbor-distinguishing totally $(\Delta +3)$-colorable. In this paper, we extend this result to the case $\Delta =9$. Namely, we prove that every planar graph G with $\Delta =9$ is neighbor-distinguishing totally 12-colorable.

Appendix

In MATLAB, the program calculating $\frac{\partial ^{k_{1}+k_{2}+\cdots +k_{n}}Q}{\partial x_{1}^{k_{1}}\partial x_{2}^{k_{2}}\cdots \partial x_{n}^{k_{n}}}$ is given as follows.