On list critical graphs

Abstract

In this paper we discuss some basic properties of k -list critical graphs. A graph G is k -list critical if there exists a list assignment L for G with | L ( v ) | = k − 1 for all vertices v of G such that every proper subgraph of G is L -colorable, but G itself is not L -colorable. This generalizes the usual definition of a k -chromatic critical graph, where L ( v ) = { 1 , … , k − 1 } for all vertices v of G . While the investigation of k -critical graphs is a well established part of coloring theory, not much is known about k -list critical graphs. Several unexpected phenomena occur, for instance a k -list critical graph may contain another one as a proper induced subgraph, with the same value of k . We also show that, for all 2 ≤ p ≤ k , there is a minimal k -list critical graph with chromatic number p . Furthermore, we discuss the question, for which values of k and n is the complete graph K n k -list critical. While this is the case for all 5 ≤ k ≤ n , K n is not 4-list critical if n is large.