Discussiones Mathematicae Graph Theory 30(1) (2010) 137-153

ON MULTISET COLORINGS OF GRAPHS

Futaba Okamoto

Mathematics Department
University of Wisconsin - La Crosse
La Crosse, WI 54601, USA

Ebrahim Salehi

Department of Mathematical Sciences
University of Nevada Las Vegas
Las Vegas, NV 89154, USA

Ping Zhang

Department of Mathematics
Western Michigan University
Kalamazoo, MI 49008, USA

Abstract

A vertex coloring of a graph G is a multiset coloring if the multisets of colors of the neighbors of every two adjacent vertices are different. The minimum k for which G has a multiset k-coloring is the multiset chromatic number χm(G) of G. For every graph G, χm(G) is bounded above by its chromatic number χ(G). The multiset chromatic numbers of regular graphs are investigated. It is shown that for every pair k, r of integers with 2 ≤ k ≤ r-1, there exists an r-regular graph with multiset chromatic number k. It is also shown that for every positive integer N, there is an r-regular graph G such that χ(G)-χm(G) = N. In particular, it is shown that χm(Kn ×K2) is asymptotically √n. In fact, χm(Kn ×K2) = χm( cor
(Kn+1)). The corona cor
(G) of a graph G is the graph obtained from G by adding, for each vertex v in G, a new vertex v′ and the edge vv′. It is shown that χm( cor
(G)) ≤ χm(G) for every nontrivial connected graph G. The multiset chromatic numbers of the corona of all complete graphs are determined.

On Multiset Colorings of Graphs

From this, it follows that for every positive integer N, there exists a graph G such that χm(G)-χm( cor
(G)) ≥ N. The result obtained on the multiset chromatic number of the corona of complete graphs is then extended to the corona of all regular complete multipartite graphs.

Keywords: vertex coloring, multiset coloring, neighbor-distinguishing coloring.

2010 Mathematics Subject Classification: 05C15.

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Received 15 November 2008
Revised 28 April 2009
Accepted 28 April 2009