Hall's condition yields less for multicolorings

It has been shown that a vertex list assignment satisfying Hall's condition is sufficient for the existence of a proper list coloring if and only if every block of the graph is a clique. It is asked in [5] whether this can be extended to multicolorings. It is shown here that in general this is not the case. Introduction The notion of list coloring has its origins independently with Vizing [7] and Erdos, Rubin & Taylor [1]. When the graph is a complete graph, list colorability with sets Ay available at each vertex v is equivalent to the set system {Av:v is a vertex} having a system of distinct representatives. Thus when the graph is complete the conditions given by P. Hall [2] are sufficient for the existence of a list coloring. Hilton and Johnson [4] noticed this and introduced the idea of placing conditions on the lists which would imply list colorability. Because of the condition's similarity to those given by Hall, Hilton and Johnson dubbed it Hall's condition. Suppose G is a finite simple graph, L:V(G)-{finite subsets of a set S} and K:V(G)-fN. G is said to be (L,IC)-colorable if there exist Av ~ L(v) with 1Av I = K( v) for every y EV(G) such that AunAv = 0 whenever uv EE(G). For each a E S and induced subgraph H of G, let a( a,L,H) denote the size of the largest independent set of vertices in H whose lists contain a. Definition. G, Land IC defined as above are said to satisfy Hall's condirion if and only if for each induced subgraph H of G ~a(a,L,H) ~ )K(V). aES VE~H) Note that Hall's condition is necessary for the existence of a proper (L,IC)-coloring. Australasian Journal of Combinatorics 18(1998), pp.263-266 Hilton and Johnson [4,6] have shown that when lC .. '1 the following extension of Hall's Theorem holds. Theorem 1. G, Land IC satisfying Hall's condition is sufficient for the existence of a proper (L,IC)-coloring when 1(" 1 if and only if every block of G is a clique. Halmos and Vaughn [3] generalized Hall's Marriage Theorem to the case where the set representatives are not just Singletons, but are subsets of a given size. In other words, they showed that the existence of a proper list multicoloring of a complete graph is merely equivalent to a generalized Hall's Marriage Theorem. We can derive Halmos and Vaughn's result from theorem 1 by taking G to be a complete graph, L:V(G)-{finite subsets of S} and lC:V(G)-IN. We obtain an auxiliary graph G* and list assignment L*:V(G*)-{finite subsets of S} by replacing each vertex VEV(G) with a KlC(v) and defining L*(u) = L(v) when uis a vertex of the KlC(v) inserted for the vertex v. We take lC* .. 1 and note that G* is also a complete graph with G*, L* and lC* satisfying Hall's condition if and only if G, L, and K satisfy Hall's condition. For a complete account of this see [5]. The Counterexample The idea used to derive Halmos and Vaughn's result when applied to graphs where every block is a clique when there is more than one block may cause cutvertices to no longer be cut-vertices. Thus we may not be able to generalize Theorem 1 in this manner to arbitrary lC:V(G)-IN. In fact, as the main intent of this exposition is to show, the general analogue to Theorem 1 does not hold. We next give an example to show that this is the case. Example 2. Let G be the graph in the figure below. Also illustrated is a list function L:V(G)-p({a,b,c,d,e,fl) and a function lC:V(G)-IN, Although G, Land lC satisfy Hall's condition, there is no proper (L,lC)-coloring of G. So the analogue to Theorem I does not exist for multicolorings in general. a1>,c .... --tIIIII---a_-..... 2