Fan's condition for completely 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.