Equitable Partitions to Spanning Trees in a Graph

Abstract

In this paper we first prove that if the edge set of an undirected graph is the disjoint union of two of its spanning trees, then for every subset $P$ of edges there exists a spanning tree decomposition that cuts $P$ into two (almost) equal parts. The main result of the paper is a further extension of this claim: If the edge set of a graph is the disjoint union of two of its spanning trees, then for every stable set of vertices of size 3, there exists such a spanning tree decomposition that cuts the stars of these vertices into (almost) equal parts. This result fails for 4 instead of 3. The proofs are elementary.