Existence Of 2-factors Containing Large Cycles In Bipartite Graphs

In this paper, we study the sufficient conditions of existence of 2-factors containing large cycles in bipartite graphs. The main results are given as follows.(1) Let G = (V₁, V₂;E) be a bipartite graph with |V₁| = |V₂|=n≥sk+1,where s≥4 and k≥1 are two integers. We define the minimum degree sum of nonadjacent vertices of graph G to beσ₂(G) = min{d(u,G) + d(v,G):u,v∈V(G), uv (?) E(G)}. ifσ₂(G)≥then G has a 2-factor with exactly k vertex-disjoint cycles of length at least 2 s.(2) Let G = (V₁, V₂;E) be a bipartite graph with |V₁| = |V₂|=n≥sk,where k≥2, s≥4, n be three integers ,ifσ_1,1 (G)≥,then for any independent edges e₁,…e_k,G contains a 2-factor with k cycles C₁,…C_k such that e_i∈E(C_i)and |C_i|≥2s.