Note on vertex-partitions of infinite graphs

Let G be a (finite or infinite) graph with vertex set V(G) and edge set E(G). A partition P = {vi: i E Z} of V(G) is called a k-partition if [Vi1 s k for every i E I. Let the graph Gp be defined by (i) V(Gp) = {vi: i E Z}, (ii) l$l$ E E(G,)exy E (G) for some x E l$, y E l$ (i #j). Given an infinite graph G, let deg,(G) denote the least d for which there exist a natural number k and a k-partition P of V(G) such that the degree of every vertex of G,, is at most d. Further, let