Fan-Type Condition for Two Completely Independent Spanning Trees

Abstract

The Spanning trees $T_1,T_2, \dots,T_k$ of a graph $G$ are called $k$ completely independent spanning trees (CISTs) if for any two vertices $u,v\in V(G)$, the paths connecting $u$ and $v$ in any two distinct trees are pairwise edge-disjoint and internally vertex-disjoint.  CISTs have significant applications in fault-tolerant broadcasting for interconnection networks, substantially improving network reliability and redundancy. However, determining whether a connected graph contains two CISTs is known to be NP-complete. Araki [J. Graph Theory, 77 (2014) 171-179.] posed an open question regarding whether certain sufficient conditions for hamiltonian cycles could ensure the existence of two CISTs. In this paper, we provide an affirmative answer to this question by proving that every connected graph $G$  of order $n\geq 7$ with $\mu_2(G)\geq n$ contains two CISTs. Notably, both the lower bounds on $n$ and the degree condition $\mu_2(G)$ are best possible.

Keywords: Completely independent spanning trees, CIST-partition, Fan-type condition