On defect restricted matching extension graphs

Abstract

Let G be a graph, v(G) the order of G, and κ(G) the connectivity of G. Let m,n,d be non-negative integers. A matching of a graph G is a set of independent edges of G and a matching is called perfect if it covers all vertices of G. A matching is called a defect-dmatching if it covers exactly v(G)−d vertices of G. A graph G with at least 2m+2n+2 vertices is said to have propertyE(m,n) if for any two disjoint matchings M,N⊆E(G) of size m,n, respectively, there is a perfect matching F such that M⊆F and N∩F=0̸. A graph G with at least 2m+2n+d+2 vertices is said to have propertyE(m,n,d) if for any two disjoint matchings M,N in G of size m,n, respectively, there is a defect-d matching F such that M⊆F and N∩F=0̸. In this paper, we generalize the concept of property E(m,n) to property E(m,n,d). We show some results of graphs with property E(m,n,d) and prove that its connectivity may be any positive integer.