On multiset colorings of graphs

• Autorzy: Futaba Okamoto , Ebrahim Salehi , Ping Zhang
• Języki publikacji: EN

Abstrakty

EN

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, $χₘ(Kₙ × K₂) = χₘ(cor(K_{n+1}))$. 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 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.

Słowa kluczowe

EN

vertex coloring; multiset coloring; neighbor-distinguishing coloring

Kategorie tematyczne

• 05C15: Coloring of graphs and hypergraphs

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2010
• Tom: 30
• Numer: 1
• Strony: 137-153

Daty

wydano

2010

otrzymano

2008-11-15

poprawiono

2009-04-28

zaakceptowano

2009-04-28

Twórcy

autor

Futaba Okamoto

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

autor

Ebrahim Salehi

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

autor

Ping Zhang

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

Bibliografia

• [1] 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.
• [2] 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.
• [3] R.L. Brooks, On coloring the nodes of a network, Proc. Cambridge Philos. Soc. 37 (1941) 194-197, doi: 10.1017/S030500410002168X.
• [4] A.C. Burris, On graphs with irregular coloring number 2, Congr. Numer. 100 (1994) 129-140.
• [5] G. Chartrand, H. Escuadro, F. Okamoto and P. Zhang, Detectable colorings of graphs, Util. Math. 69 (2006) 13-32.
• [6] 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.
• [7] G. Chartrand, F. Okamoto, E. Salehi and P. Zhang, The multiset chromatic number of a graph, Math. Bohem. 134 (2009) 191-209.
• [8] G. Chartrand and P. Zhang, Chromatic Graph Theory (Chapman & Hall/CRC Press, Boca Raton, FL, 2009).
• [9] H. Escuadro, F. Okamoto and P. Zhang, A three-color problem in graph theory, Bull. Inst. Combin. Appl. 52 (2008) 65-82.
• [10] 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.
• [11] M. Radcliffe and P. Zhang, Irregular colorings of graphs, Bull. Inst. Combin. Appl. 49 (2007) 41-59.

Identyfikatory

DOI

10.7151/dmgt.1483