Construction and Conditions for Completely Independent Spanning Trees in Hypercubes and Regular Bipartite Graphs

Abstract

A set of \( k \) spanning trees in a graph \( G \) is called a set of \textit{completely independent spanning trees (CISTs)} if, for every pair of vertices \( x \) and \( y \), the paths connecting \( x \) and \( y \) across different trees do not share any vertices or edges, except for \( x \) and \( y \) themselves. Hasunuma conjectured that every \(2k\)-connected graph contains exactly \(k\) completely independent spanning trees (CISTs). However, P\'et\'erfalvi disproved this conjecture. When \( k = 2 \), the two CISTs are called a \textit{dual-CIST}. It has been shown that determining whether a graph can have \( k \) CISTs is an NP-complete problem, even when \( k = 2 \). In $2017$, Darties et al. raised the question of whether the $6-$dimensional hypercube \( Q_6 \) can have three completely independent spanning trees (CISTs). This paper provides an answer to that question. In this paper, we first present a necessary condition for \( k \)-regular, \( k \)-connected bipartite graphs to have \( \left\lfloor \frac{k}{2} \right\rfloor \) CISTs. We also investigate that the hypercube of dimension \( n \) cannot have \( \frac{n}{2} \) CISTs, which means Hasunuma's conjecture does not hold for the hypercube \( Q_n \) when \( n \) is an even integer \(2 < n \leq 10^7 \), except when \(n = 2^r\) and \( n \in \{161038, 215326, 2568226, 3020626, 7866046, 9115426 \} \). This result also resolves a question posed by Darties et al. The construction of multiple CISTs on the underlying graph of a network has practical applications in ensuring the fault tolerance of data transmission. In this context, we also provide a construction for three completely independent spanning trees in the hypercube \(Q_n\) for \(n \geq 7\). Our results show that Hasunuma's conjecture holds for odd integer \(n = 7\) in \(Q_n\), but does not hold for even integer \(n = 6\).