Journal of Combinatorial Optimization

, Volume 27, Issue 2, pp 379–396

The adjacent vertex distinguishing total coloring of planar graphs

Article

DOI: 10.1007/s10878-012-9527-2

Cite this article as:
Wang, W. & Huang, D. J Comb Optim (2014) 27: 379. doi:10.1007/s10878-012-9527-2

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.

Keywords

Adjacent vertex distinguishing total coloringPlanar graphMaximum degree

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsZhejiang Normal UniversityJinhuaChina