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Discussiones Mathematicae Graph Theory 30(1) (2010) 137-153
DOI: https://doi.org/10.7151/dmgt.1483

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 χ_m(G) of G. For every graph G, χ_m(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)-χ_m(G) = N. In particular, it is shown that χ_m(K_n ×K₂) is asymptotically √n. In fact, χ_m(K_n ×K₂) = χ_m( cor
(K_n+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 χ_m( cor
(G)) ≤ χ_m(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 χ_m(G)-χ_m( 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