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## Discussiones Mathematicae Graph Theory

2010 | 30 | 1 | 137-153
Tytuł artykułu

### On multiset colorings of graphs

Autorzy
Treść / Zawartość
Warianty tytułu
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
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
137-153
Opis fizyczny
Daty
wydano
2010
otrzymano
2008-11-15
poprawiono
2009-04-28
zaakceptowano
2009-04-28
Twórcy
autor
• Mathematics Department, University of Wisconsin - La Crosse, La Crosse, WI 54601, USA
autor
• Department of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, NV 89154, USA
autor
• 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.
Typ dokumentu
Bibliografia
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