Partitioning infinite trees

A graph G is bisectable if its edges can be colored by two colors so that the resulting monochromatic subgraphs are isomorphic. We show that any infinite tree of maximum degree Delta with infinitely many vertices of degree at least Delta - 1 is bisectable as is any infinite tree of maximum degree Delta less than or equal to 14. Further, it is proved that every infinite tree T of finite maximum degree contains a finite subset E of its edges so that the graph T-E is bisectable. To measure how "far" a graph G is from being bisectable, we define c(G) to be the smallest number k>1 so that there is a coloring of the edges of G by k colors with the property that any two monochromatic subgraphs are isomorphic. An upper bound on c(G), which is in a sense best possible, is presented. (C) 2000 John Wiley & Sons, Inc.