Two completely independent spanning trees of claw-free graphs

Abstract

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}. If the subgraph induced by the neighbor set of a vertex u in a graph G is connected, then u is an eligible vertex of G. An hourglass is a graph with degree 4,2,2,2,2 and (P6)2 denotes the square graph of a path P6 on six vertices. Araki (2014) [1] proved that if a graph G satisfies the Dirac's condition, which is also sufficient condition for hamiltonian graphs, then G contains two CISTs. Ryjáček proposed that a claw-free graph G is hamiltonian if and only if its Ryjáček's closure, denoted by cl(G), is hamiltonian. In this paper, we prove that if G is a {claw, hourglass, (P6)2}-free graph with δ(G)≥4, then G contains two CISTs if and only if cl(G) has two CISTs. The bound of the minimum degree in our result is best possible.