Author(s)

Wenduan Dai, Yan Liu^*, Yanfang Wu

School of Mathematical Sciences, South China Normal University, Guangzhou, China.

Let k be a positive integer and G a bipartite graph with bipartition $\left(X,Y\right)$ . 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 $\left|X\right|-d$ 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.

Bipartite Graph, Semi-Matching, Perfect 1-k Matching, k-Elementary Graph

Dai, W. , Liu, Y. and Wu, Y. (2024) Perfect 1-k Matchings of Bipartite Graphs. Open Journal of Discrete Mathematics, 14, 43-53. doi: 10.4236/ojdm.2024.144005.