Perfect 1- k Matchings of Bipartite Graphs
- 1 School of Mathematical Sciences, South China Normal University, Guangzhou, China
- 2 School of Mathematical Sciences, South China Normal University, Guangzhou, China
- 3 School of Mathematical Sciences, South China Normal University, Guangzhou, China
Abstract
Let k be a positive integer and G a bipartite graph with bipartition ( X , Y ) . A perfect 1- k matching is an edge subset M of G such that each vertex in Y is incident with exactly one edge in M and each vertex in X is incident with exactly k edges in M . A perfect 1- k matching is an optimal semi-matching related to the load-balancing problem, where a semi-matching is an edge subset M such that each vertex in Y is incident with exactly one edge in M , and a vertex in X can be incident with an arbitrary number of edges in M . In this paper, we give three sufficient and necessary conditions for the existence of perfect 1- k matchings and for the existence of 1- k matchings covering | X | − d vertices in X , respectively, and characterize k -elementary bipartite graph which is a graph such that the subgraph induced by all k -allowed edges is connected, where an edge is k -allowed if it is contained in a perfect 1- k matching.
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