Comments on “A Hamilton sufficient condition for completely independent spanning tree”

Abstract

Spanning trees T1,T2,…,Tk (k≥2) in a graph G are called completely independent spanning trees (CISTs for short) if for any two vertices x,y of G, the paths joining x and y in these k trees are pairwise openly disjoint. Hong and Zhang (2018) recently showed that a sufficient condition for Hamiltonian graphs still suffices for the existence of two CISTs. That is, if G is a graph with n vertices and |N(x)∪N(y)|≥n2, |N(x)∩N(y)|≥3 for every two nonadjacent vertices x,y of G and n≥5, then G admits two CISTs. In this note, we first attend that the restriction on the number of vertices in the statement should be revised. Moreover, we point out that there is a flaw in their proof. Accordingly, we give an amendment to correct the proof.