Isolated toughness and P>3-factors in graphs

Abstract

Let m,n,k be three positive integers. A path factor with each component having at least n vertices is called a P>n-factor. A graph G is defined as a (P> m)-factor deleted graph if G-E' has a P>n-factor for every E' C E(G) with E = m. A graph G is defined as a (P>n, fc)-factor critical graph if G-U admits a P>n-factor for every U C V(G) with U = k. In this paper, we verify the following two results: (1) A graph G with n(G) > 2ia±i is a (P>3,m)-factor deleted graph if 1(G) > ; (2) A graph G with k(G) > k + 2 is a (P>3,k)-iactox critical graph if 1(G) > Furthermore, some conditions in main results are sharp. © 2020 Utilitas Mathematica Publishing Inc.. All rights reserved.