Characterization of graphs with Hall number 2

Hall's condition is a simple requirement that a graph G and list assignment L must satisfy if G is to have a proper L-colouring. The Hall number of G is the smallest integer m such that whenever the lists on the vertices each has size at least m and Hall's condition is satisfied a proper L-colouring exists. Hilton and P.D. Johnson introduced the parameter and showed that a graph has Hall number 1 if and only if every block is a clique. In this paper we give a forbidden-induced-subgraph, characterization of graphs with Hall number 2. (C) 2003 Wiley Periodicals, Inc. J Graph Theory 45: 81-100, 2004.