TITLE:
Perfect 1-k Matchings of Bipartite Graphs
AUTHORS:
Wenduan Dai, Yan Liu, Yanfang Wu
KEYWORDS:
Bipartite Graph, Semi-Matching, Perfect 1-k Matching, k-Elementary Graph
JOURNAL NAME:
Open Journal of Discrete Mathematics,
Vol.14 No.4,
September
25,
2024
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.