Construction of Completely Independent Spanning Tree Based on Vertex Degree

Abstract

Interconnection networks have been extensively studied in the field of parallel computer systems. In the interconnection network, completely independent spanning tree (CISTs) plays an important role in the reliable transmission, parallel transmission, and safe distribution of information. Two spanning trees $T_1$ and $T_2$ of graph G are completely independent if, for any two distinct vertices u and v of G, the two paths from u to v on $T_1$ and $T_2$ are internally disjoint. The spanning trees $T_1, T_2, \ldots , T_k$ of G are completely independent spanning trees if they are pairwise completely independent. In 2015, Hasunuma proof that G has $\lfloor \frac{n(G)}{k}\rfloor$ CISTs if $\delta (G) \ge n(G) - k$, $3 \le k \le \frac{n(G)}{2}$ and $n(G) \ge 7$. In this paper, we prove that G has $\lfloor \frac{5n(G)}{12} \rfloor$ CISTs if $\delta (G) \ge n(G)-2$ and $n(G) \ge 12$, and G has $(t+1)$-CISTs if $\delta (G) \ge n(G)-3$ and $n(G) = 3t - 2 \ge 23$.