Two completely independent spanning trees of split graphs

Abstract

G is a split graph if V(G) can be partitioned into a clique D and an independent set I. Let τ(G) denote the toughness of a graph G. If a graph G contains two spanning trees T1,T2 such that for each two distinct vertices x,y of G, the (x,y)-path in each Ti has no common edge and no common vertex except for the two ends, then T1,T2 are called two completely independent spanning trees (CISTs) of G,i∈{1,2}. Several results have shown that some sufficient conditions for Hamiltonian graphs also guarantee the existence of two CISTs, which implies the Hamiltonicity and the existence of two CISTs of a graph have close connection. In this paper, we prove that if G is a Hamiltonian split graph with |D|>max{3,|I|}, then G contains two CISTs. Moreover, we show that if G is a Hamiltonian split graph with τ(G)>1, then G contains two CISTs. As a corollary, we obtain that every 32-tough split graph contains two CISTs.