Proof of a Conjecture About Minimum Spanning Tree Cycle Intersection.

Let G be a graph and T a spanning tree of G. For an edge e in G − T , there is a cycle in T ∪ { e }. We call those edges cycle-edges and those cycles tree-cycles. The intersection of two tree-cycles is the set of all edges in common. If the intersection of two distinct tree-cycles is not empty, we regard that as an intersection. The tree intersection number of T is the number of intersections among all tree-cycles of T. In this paper, we prove the conjecture, posed by Dubinsky et al. (2021), which states that if a graph admits a star spanning tree in which one vertex is adjacent to all other vertices, then the star spanning tree has the minimum tree intersection number. [ABSTRACT FROM AUTHOR]