k-regular factors and semi-k-regular factors in bipartite graphs

Abstract

LetG be a graph, andk≥1 an integer. LetU be a subset ofV(G), and letF be a spanning subgraph ofG such that deg _F (x)=k for allx ∈V(G)−U. If deg _F (x)≥k for allx∈U, thenF is called an upper semi-k-regular factor with defect setU, and if deg _F (x)≤k for allx∈U, thenF is called a lower semi-k-regular factor with defect setU. Now letG=(X, Y;E(G)) be a bipartite graph with bipartition (X,Y) such that ∣X∣=∣Y∣≥k+2. We prove the following two results.

(1) Suppose that for each subsetU ₁⊂X such that ∣U ₁∣=max{k+1, ⌈∣X∣+1/2⌉},G has an upper semi-k-regular factor with defect setU ₁∪Y, and for each subsetU ₂⊂Y such that ∣U ₂∣=max{k+1, ⌈∣X∣+1/2⌉},G has an upper semi-k-regular factor with defect setX∪U ₂. ThenG has ak-factor.

(2) Suppose that for each subsetU ₁⊂X such that ∣U ₁∣=∈∣X∣−1/k+1∉,G has a lower semi-k-regular factor with defect setU ₁∪Y, and for each subsetU ₂⊂Y such that ∣U ₂∣=∈∣X∣−1/k+1∉,G has a lower semi-k-regular factor with defect setX∪U ₂. ThenG has ak-factor.