Perfect integer k-matching, k-factor-critical, and the spectral radius of graphs.

A graph G is k -factor-critical if G − S has a perfect matching for any subset S of V (G) with | S | = k. An integer k -matching of G is a function h : E (G) → { 0 , 1 , ... , k } satisfying ∑ e ∈ Γ (v) h (e) ≤ k for all v ∈ V (G) , where Γ (v) is the set of edges incident with v. An integer k -matching h of G is called perfect if ∑ e ∈ E (G) h (e) = k | V (G) | / 2. A graph G has the strong parity property if for every subset S of V (G) with even size, G has a spanning subgraph F with minimum degree at least one such that d F (v) ≡ 1 (mod 2) for all v ∈ S and d F (u) ≡ 0 (mod 2) for all u ∈ V (G) ﹨ S. In this paper, we provide edge number and spectral conditions for the k -factor-criticality, perfect integer k -matching and strong parity property of a graph, respectively. [ABSTRACT FROM AUTHOR]