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ISSN 2083-5892 (electronic version)

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory


Discussiones Mathematicae Graph Theory 33(1) (2013) 231-242
DOI: 10.7151/dmgt.1660

Dedicated to Mietek Borowiecki on the occasion of his seventieth birthday

Choice-perfect Graphs

Zsolt Tuza

Alfréd Rényi Institute of Mathematics
Hungarian Academy of Sciences
H-1053 Budapest, Reáltanoda u. 13-15, Hungary
Department of Computer Science and Systems Technology
University of Pannonia
H-8200 Veszprém, Egyetem u. 10, Hungary


Given a graph G = (V,E) and a set Lv of admissible colors for each vertex v ∈ V (termed the list at v), a list coloring of G is a (proper) vertex coloring φ:V → ∪v ∈ V Lv such that φ(v) ∈ Lv for all v ∈ V and φ(u) ≠ φ(v) for all uv ∈ E. If such a φ exists, G is said to be list colorable. The choice number of G is the smallest natural number k for which G is list colorable whenever each list contains at least k colors.

In this note we initiate the study of graphs in which the choice number equals the clique number or the chromatic number in every induced subgraph. We call them choice-ω-perfect and choice-χ-perfect graphs, respectively. The main result of the paper states that the square of every cycle is choice-χ-perfect.

Keywords: graph coloring, list coloring, choice-perfect graph

2010 Mathematics Subject Classification: 05C15, 05C17, 05C75.


[1]N. Alon and M. Tarsi, Colorings and orientations of graphs, Combinatorica 12 (1992) 125--134, doi: 10.1007/BF01204715.
[2]M. Borowiecki, I. Broere, M. Frick, P. Mihók and G. Semanišin, A survey of hereditary properties of graphs, Discuss. Math. Graph Theory 17 (1997) 5--50, doi: 10.7151/dmgt.1037.
[3]M. Borowiecki, E. Sidorowicz and Zs. Tuza, Game list colouring of graphs, Electron. J. Combin. 14 (2007) #R26.
[4]P. Erdös, A.L. Rubin and H. Taylor, Choosability in graphs, West-Coast Conference on Combinatorics, Graph Theory and Computing, Arcata, California, Congr. Numer. XXVI (1979) 125--157.
[5]H. Fleischner and M. Stiebitz, A solution to a colouring problem of P. Erdös, Discrete Math. 101 (1992) 39--48, doi: 10.1016/0012-365X(92)90588-7.
[6]F. Galvin, The list chromatic index of a bipartite multigraph, J. Combin. Theory (B) 63 (1995) 153--158, doi: 10.1006/jctb.1995.1011.
[7]S. Gravier and F. Maffray, Graphs whose choice number is equal to their chromatic number, J. Graph Theory 27 (1998) 87--97, doi: 10.1002/(SICI)1097-0118(199802)27:2<87::AID-JGT4>3.0.CO;2-B.
[8]S. Gravier and F. Maffray, On the choice number of claw-free perfect graphs, Discrete Math. 276 (2004) 211--218, doi: 10.1016/S0012-365X(03)00292-9.
[9]A.J.W. Hilton and P.D. Johnson, Jr., Extending Hall's theorem, in: Topics in Combinatorics and Graph Theory---Essays in Honour of Gerhard Ringel (R. Bodendiek et al., Eds.), (Teubner, 1990) 359--371.
[10]A.J.W. Hilton and P.D. Johnson, Jr., The Hall number, the Hall index, and the total Hall number of a graph, Discrete Appl. Math. 94 (1999) 227--245, doi: 10.1016/S0166-218X(99)00023-2.
[11]M. Juvan, B. Mohar and R. Škrekovski, List total colourings of graphs, Combin. Probab. Comput. 7 (1998) 181--188, doi: 10.1017/S0963548397003210.
[12]M. Juvan, B. Mohar and R. Thomas, List edge-colorings of series-parallel graphs, Electron. J. Combin. 6 (1999) #R42.
[13]D. Peterson and D.R. Woodall, Edge-choosability in line-perfect multigraphs, Discrete Math. 202 (1999) 191--199, doi: 10.1016/S0012-365X(98)00293-3.
[14]Zs. Tuza, Graph colorings with local constraints---A survey, Discuss. Math. Graph Theory 17 (1997) 161--228, doi: 10.7151/dmgt.1049.
[15]Zs. Tuza, Choice-perfect graphs and Hall numbers, manuscript, 1997.
[16]Zs. Tuza, Extremal jumps of the Hall number, Electron. Notes Discrete Math. 28 (2007) 83--89, doi: 10.1016/j.endm.2007.01.012.
[17]Zs. Tuza, Hall number for list colorings of graphs: Extremal results, Discrete Math. 310 (2010) 461--470, doi: 10.1016/j.disc.2009.03.025.
[18]Zs. Tuza and M. Voigt, List colorings and reducibility, Discrete Appl. Math. 79 (1997) 247--256, doi: 10.1016/S0166-218X(97)00046-2.
[19]V.G. Vizing, Coloring the vertices of a graph in prescribed colors, Metody Diskret. Anal. v Teorii Kodov i Schem 29 (1976) 3--10 (in Russian).
[20]D.R. Woodall, Edge-choosability of multicircuits, Discrete Math. 202 (1999) 271--277, doi: 10.1016/S0012-365X(98)00297-0.

Received 28 April 2012
Revised 5 September 2012
Accepted 5 September 2012