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 = V₀ {smile} V₁ such that every vertex of V_i is joined to at least as many vertices in V_1-i as to vertices in V_i. It is easily seen that every finite 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 m₀ < m₁ < ... < m_k such that m_i is regular for 1 ≤ i ≤ k, there are fewer than m₀ vertices having finite degrees, and every vertex having infinite degree has degree m_i for some i ≤ k.