Fan's condition for completely independent spanning trees

TL;DR

This paper proves that a certain degree sum condition in connected graphs guarantees the existence of two completely independent spanning trees, addressing a question related to Fan-type conditions and spanning tree independence.

Contribution

It establishes a sufficient condition based on degree sums for the existence of two completely independent spanning trees in connected graphs.

Findings

Graphs with degree sum at least |V(G)| for vertices at distance 2 have two independent spanning trees.

The result confirms Fan-type conditions ensure independent spanning trees.

Addresses an open question about conditions guaranteeing independent spanning trees.

Abstract

Spanning trees $T_1,T_2, \dots,T_k$ of $G$ are $k$ completely independent spanning trees if, for any two vertices $u,v\in V(G)$, the paths from $u$ to $v$ in these $k$ trees are pairwise edge-disjoint and internal vertex-disjoint. Hasunuma proved that determining whether a graph contains $k$ completely independent spanning trees is NP-complete, even for $k = 2$. Araki posed the question of whether certain known sufficient conditions for hamiltonian cycles are also also guarantee two completely independent spanning trees? In this paper, we affirmatively answer this question for the Fan-type condition. Precisely, we proved that if $G$ is a connected graph such that each pair of vertices at distance 2 has degree sum at least $|V(G)|$, then $G$ has two completely independent spanning trees.