Journal of Combinatorial Optimization

, Volume 33, Issue 2, pp 779–790

# The adjacent vertex distinguishing total coloring of planar graphs without adjacent 4-cycles

• Lin Sun
• Xiaohan Cheng
• Jianliang Wu
Article

## Abstract

A total [k]-coloring of a graph G is a mapping $$\phi$$: $$V(G)\cup E(G)\rightarrow [k]=\{1, 2,\ldots , k\}$$ such that no two adjacent or incident elements in $$V(G)\cup E(G)$$ receive the same color. In a total [k]-coloring $$\phi$$ of G, let $$C_{\phi }(v)$$ denote the set of colors of the edges incident to v and the color of v. If for each edge uv, $$C_{\phi }(u)\ne C_{\phi }(v)$$, we call such a total [k]-coloring an adjacent vertex distinguishing total coloring of G. $$\chi ''_{a}(G)$$ denotes the smallest value k in such a coloring of G. In this paper, by using the Combinatorial Nullstellensatz and the discharging method, we prove that if a planar graph G with maximum degree $$\Delta \ge 8$$ contains no adjacent 4-cycles, then $$\chi ''_{a}(G)\le \Delta +3$$.

## Keywords

Planar graph Adjacent vertex distinguishing total coloring Combinatorial Nullstellensatz Discharging method

05C15

## Notes

### Acknowledgments

This work is partially supported by NSFC (11271006).

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