Remarks on path factors in graphs

Abstract

A spanning subgraph of a graph is defined as a path factor of the graph if its component are paths. A P≥n-factor means a path factor with each component having at least n vertices. A graph G is defined as a (P≥n, m)-factor deleted graph if G–E′ has a P≥n-factor for every E′ ⊆ E(G) with |E′| = m. A graph G is defined as a (P≥n, k)-factor critical graph if after deleting any k vertices of G the remaining graph of G admits a P≥n-factor. In this paper, we demonstrate that (i) a graph G is (P≥3, m)-factor deleted if κ(G) ≥ 2m + 1 and bind(G) ≥  2/3 - $ \frac{3}{2}-\frac{1}{4m+4}$; (ii) a graph G is (P≥3, k)-factor critical if κ(G) ≥ k + 2 and bind(G) ≥ $ \frac{5+k}{4}$.