Component factors and degree sum conditions in graphs

For a set A of connected graphs, an A -factor is a spanning subgraph of a graph, whose connected components are isomorphic to graphs from the set A . An A -factor is also referred as a component factor. A graph G is called an (A, m)-factor deleted graph if for every E0 ⊆ E(G) with |E0| = m,G−E0 admits an A -factor. A graph G is called an (A, l)-factor critical graph if for every V0 ⊆V(G) with |V0| = l,G−V0 admits an A -factor. Let m,l and k be three positive integers with k ≥ 2, and write F = {P2,C3,P5,T (3)} and H = {K1,1,K1,2,...,K1,k,T (2k+1)}, where T (3) and T (2k+1) are two special families of trees. Inspired by finding a sufficient condition to check for the existence of path-factors with some special restraints, we focus on the sufficient conditions based on a graphic parameter called degree sum: σk(G)=min_{X\subseteq V(G)}{[[EQUATION]]dG(x): X is an independent set of k vertices}. In this article, we verify that (i) an (l+2)-connected graph G of order n is an (\mathcal{F},l)-factor critical graph if σ3(G)\geq\frac{6n+9l}{5}; (ii) a (2m+1)-connected graph G of order n is an (\mathcal{F},m)-factor deleted graph if σm+2(G)\geq\frac{6}{5}n; (iii) an (l+2)-connected graph G of order n is an (\mathcal{H},l)-factor critical graph if σ2k+1(G)\geq\frac{6n+(6k+3)l}{2k+3}; (iv) a (2m+1)-connected graph G of order n is an (\mathcal{H},m)-factor deleted graph if σm+2(G)\geq\frac{6n}{2k+3}.