%0 Journal Article %T Perfect 1-k Matchings of Bipartite Graphs %A Wenduan Dai %A Yan Liu %A Yanfang Wu %J Open Journal of Discrete Mathematics %P 43-53 %@ 2161-7643 %D 2024 %I Scientific Research Publishing %R 10.4236/ojdm.2024.144005 %X 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. %K Bipartite Graph %K Semi-Matching %K Perfect 1-k Matching %K k-Elementary Graph %U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=136214