Unfriendly partitions of a graph

It has been conjectured by Cowan and Emerson [3] that every graph has an unfriendly partition; i.e., there is a partition of the vertex set V= V, v V, such that every vertex of V, is joined to at least as many vertices in V, _, as to vertices in V,. It is easily seen that every rinite graph has such a partition, and hence by compactness so does any locally finite graph. We show that the conjecture is also true for graphs which satisfy one of the following two conditions: (i) there are only finitely many vertices having infinite degrees; (ii) there are a finite number of infinite cardinals “to < ntI < cm, such that m, is regular for 1 < i 6 X-, there are fewer than m,, vertices having finite degrees, and every vertex having infinite degree has degree m, for some i < k. 1’ 1990 Academx Press. Inc.