Fan's condition for completely independent spanning trees

TL;DR

This work addresses the problem of when a graph admits two completely independent spanning trees (CISTs). It leverages Araki's cist-partition framework and a detailed case analysis under the Fan-type distance-2 degree-sum condition, establishing that if $mu_2(G)\ge |V(G)|$ for a connected graph $G$, then $G$ contains two CISTs (with the analysis ensuring 2-connectivity and appropriate cuts). The authors provide a constructive proof by partitioning the vertex set around two distance-2 vertices and proving the existence of a $2$-CIST-partition, thereby guaranteeing two CISTs. They also discuss the sharpness of the bound and relate the result to Hamiltonian-type conditions, highlighting the practical impact for network reliability and redundancy.

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.

Figures (3)

• Figure 1: The illustration of a 2-CIST-partition in Lemma \ref{['lem3']}.
• Figure 2: The illustration of a 2-CIST-partition in Subcase 2.2.1.
• Figure 3: The illustration of a 2-CIST-partition in Subcase 2.2.2.

Theorems & Definitions (23)

• Definition 1.1
• Definition 1.2
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Corollary 1.1
• Theorem 1.7