# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

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# Discussiones Mathematicae Graph Theory

## 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.

## References

  L. Addario-Berry, R.E.L. Aldred, K. Dalal and B.A. Reed, Vertex colouring edge partitions, J. Combin. Theory (B) 94 (2005) 237-244, doi: 10.1016/j.jctb.2005.01.001.  M. Anderson, C. Barrientos, R.C. Brigham, J.R. Carrington, M. Kronman, R.P. Vitray and J. Yellen, Irregular colorings of some graph classes, Bull. Inst. Combin. Appl., to appear.  R.L. Brooks, On coloring the nodes of a network, Proc. Cambridge Philos. Soc. 37 (1941) 194-197, doi: 10.1017/S030500410002168X.  A.C. Burris, On graphs with irregular coloring number 2, Congr. Numer. 100 (1994) 129-140.  G. Chartrand, H. Escuadro, F. Okamoto and P. Zhang, Detectable colorings of graphs, Util. Math. 69 (2006) 13-32.  G. Chartrand, L. Lesniak, D.W. VanderJagt and P. Zhang, Recognizable colorings of graphs, Discuss. Math. Graph Theory 28 (2008) 35-57, doi: 10.7151/dmgt.1390.  G. Chartrand, F. Okamoto, E. Salehi and P. Zhang, The multiset chromatic number of a graph, Math. Bohem. 134 (2009) 191-209.  G. Chartrand and P. Zhang, Chromatic Graph Theory (Chapman & Hall/CRC Press, Boca Raton, FL, 2009).  H. Escuadro, F. Okamoto and P. Zhang, A three-color problem in graph theory, Bull. Inst. Combin. Appl. 52 (2008) 65-82.  M. Karoński, T. Łuczak and A. Thomason, Edge weights and vertex colours, J. Combin. Theory (B) 91 (2004) 151-157, doi: 10.1016/j.jctb.2003.12.001.  M. Radcliffe and P. Zhang, Irregular colorings of graphs, Bull. Inst. Combin. Appl. 49 (2007) 41-59.