, Volume 27, Issue 2, pp 379-396
Date: 06 Jul 2012

The adjacent vertex distinguishing total coloring of planar graphs

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Abstract

An adjacent vertex distinguishing total coloring of a graph G is a proper total coloring of G such that any pair of adjacent vertices have distinct sets of colors. The minimum number of colors needed for an adjacent vertex distinguishing total coloring of G is denoted by \(\chi''_{a}(G)\) .

In this paper, we characterize completely the adjacent vertex distinguishing total chromatic number of planar graphs G with large maximum degree Δ by showing that if Δ≥14, then \(\varDelta+1\leq \chi''_{a}(G)\leq \varDelta+2\) , and \(\chi''_{a}(G)=\varDelta+2\) if and only if G contains two adjacent vertices of maximum degree.

First author is supported partially by NSFC (No. 11071223), ZJNSFC (No. Z6090150), ZJIP (No. T200905), ZSDZZZZXK13 and IP-OCNS-ZJNU. Second author is supported by the Foundation of Zhejiang Educational Committee (No. Y201226078).