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 and Department of Computer Science and Systems Technology University of Pannonia H-8200 Veszprém, Egyetem u. 10, Hungary e-mail: tuza@dcs.uni-pannon.hu

## Abstract

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.

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