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## Discussiones Mathematicae Graph Theory

2012 | 32 | 4 | 795-806
Tytuł artykułu

### The s-packing chromatic number of a graph

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let S = (a₁, a₂, ...) be an infinite nondecreasing sequence of positive integers. An S-packing k-coloring of a graph G is a mapping from V(G) to {1,2,...,k} such that vertices with color i have pairwise distance greater than $a_i$, and the S-packing chromatic number $χ_S(G)$ of G is the smallest integer k such that G has an S-packing k-coloring. This concept generalizes the concept of proper coloring (when S = (1,1,1,...)) and broadcast coloring (when S = (1,2,3,4,...)). In this paper, we consider bounds on the parameter and its relationship with other parameters. We characterize the graphs with $χ_S = 2$ and determine $χ_S$ for several common families of graphs. We examine $χ_S$ for the infinite path and give some exact values and asymptotic bounds. Finally we consider complexity questions, especially about recognizing graphs with $χ_S = 3$.
Słowa kluczowe
EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
795-806
Opis fizyczny
Daty
wydano
2012
otrzymano
2011-08-27
poprawiono
2012-02-22
zaakceptowano
2012-02-23
Twórcy
autor
• Dept of Mathematical Sciences, Clemson University, Clemson SC 29634
autor
• Dept of Mathematical Sciences, Clemson University, Clemson SC 29634
Bibliografia
• [1] B. Brešar and S. Klavžar and D.F. Rall, On the packing chromatic number of Cartesian products, hexagonal lattice, and trees, Discrete Appl. Math. 155 (2007) 2303-2311, doi: 10.1016/j.dam.2007.06.008.
• [2] J. Ekstein, J.Fiala, P.Holub and B. Lidický, The packing chromatic number of the square lattice is at least 12, preprint.
• [3] J. Fiala and P.A. Golovach, Complexity of the packing coloring problem for trees, Discrete Appl. Math. 158 (2010) 771-778, doi: 10.1016/j.dam.2008.09.001.
• [4] J. Fiala, S. Klavžar and B. Lidický, The packing chromatic number of infinite product graphs, European J. Combin. 30 (2009) 1101-1113, doi: 10.1016/j.ejc.2008.09.014.
• [5] M.R. Garey and D.S. Johnson, Computers and Intractability, A guide to the Theory of NP-completeness (W. H. Freeman and Co., San Francisco, Calif., 1979).
• [6] W. Goddard, S.M. Hedetniemi, S.T. Hedetniemi, J.M. Harris and D.F. Rall, Broadcast chromatic numbers of graphs, Ars Combin. 8 (2008) 33-49.
• [7] C. Sloper, An eccentric coloring of trees, Australas. J. Combin. 29 (2004) 309-321.
• [8] R. Soukal and P. Holub, A note on packing chromatic number of the square lattice, Electron. J. Combin. 17 (2010) Note 17, 7.
• [9] D.B. West, Introduction to Graph Theory (Prentice Hall, NJ, USA, 2001).
Typ dokumentu
Bibliografia
Identyfikatory