Adjacent vertex distinguishing  edge-colorings and total-colorings  of   the Cartesian product of graphs

Shuangliang Tian; Ping Chen; Yabin Shao; Qian Wang

doi:10.3934/naco.2014.4.49

Article Contents

2014, Volume 4, Issue 1: 49-58. Doi: 10.3934/naco.2014.4.49

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Adjacent vertex distinguishing edge-colorings and total-colorings of the Cartesian product of graphs

1.
College of Mathematics and Computer Science, Northwest University for Nationalities, Lanzhou 730030
2.
College of Management, Northwest University for Nationalities, Lanzhou 730030

Received: February 28, 2013

Revised: October 31, 2013

Published: November 2013

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Abstract

Let $G$ be a simple graph with vertex set $V(G)$ and edge set $E(G)$. An edge-coloring $\sigma$ of $G$ is called an adjacent vertex distinguishing edge-coloring of $G$ if $F_{\sigma}(u)\not= F_{\sigma}(v)$ for any $uv\in E(G)$, where $F_{\sigma}(u)$ denotes the set of colors of edges incident with $u$. A total-coloring $\sigma$ of $G$ is called an adjacent vertex distinguishing total-coloring of $G$ if $S_{\sigma}(u)\not= S_{\sigma}(v)$ for any $uv\in E(G)$, where $S_{\sigma}(u)$ denotes the set of colors of edges incident with $u$ together with the color assigned to $u$. The minimum number of colors required for an adjacent vertex distinguishing edge-coloring (resp. an adjacent vertex distinguishing total-coloring) of $G$ is denoted by $\chi_a^{'}(G)$ (resp. $\chi^{''}_{a}(G)$). In this paper, we provide upper bounds for these parameters of the Cartesian product $G$ □ $H$ of two graphs $G$ and $H$. We also determine exact value of these parameters for the Cartesian product of a bipartite graph and a complete graph or a cycle, the Cartesian product of a complete graph and a cycle, the Cartesian product of two trees and the Cartesian product of regular graphs.

Keywords:

Edge-coloring,
total-coloring,
adjacent vertex distinguishing edge-coloring,
adjacent vertex distinguishing total-coloring,
Cartesian product.

Mathematics Subject Classification: Primary: 05C35, 05C75; Secondary: 05C60.

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References

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