Independent triangles covering given vertices of a graph

Abstract

Let G be a simple graph of order n, k a positive integer with n⩾3k and X a set of any k vertices of G. We show that if the minimum degree δ(G)⩾(n+k)/2, then G contains k independent triangles covering all vertices of X; and if the minimum degree δ(G)⩾(n+2k)/2, then G contains k independent triangles such that each triangle contains exactly one vertex of X. The bounds on the minimum degree of G in above results are sharp. Some conjectures about independent triangles covering some given vertices are proposed.