On Rainbow-Cycle-Forbidding Edge Colorings of Finite Graphs

Abstract

It is shown that whenever the edges of a connected simple graph on n vertices are colored with $$n-1$$ colors appearing so that no cycle in G is rainbow, there must be a monochromatic edge cut in G. From this it follows that such colorings of G can be represented, or ‘encoded,’ by full binary trees with n leaves, with vertices labeled by subsets of V(G), such that the leaf labels are singletons, the label of each non-leaf is the union of the labels of its children, and each label set induces a connected subgraph of G. It is also shown that $$n-1$$ is the largest integer for which the main theorem holds, for each n, although for some graphs a certain strengthening of the hypothesis makes the theorem conclusion true with $$n-1$$ replaced by $$n-2$$.