Construction and Conditions for Completely Independent Spanning Trees in Hypercubes and Regular Bipartite Graphs (2410.03379v2)

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).