Equimatchable Bipartite Graphs

IF 0.5 4区数学 Q3 MATHEMATICS

Discussiones Mathematicae Graph Theory Pub Date : 2022-11-25 DOI:10.7151/dmgt.2356

Yasemin Büyükçolak, Didem Gözüpek, Sibel Ozkan

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引用次数: 0

Abstract

Abstract A graph is called equimatchable if all of its maximal matchings have the same size. Lesk et al. [Equi-matchable graphs, Graph Theory and Combinatorics (Academic Press, London, 1984) 239–254] has provided a characterization of equimatchable bipartite graphs. Motivated by the fact that this characterization is not structural, Frendrup et al. [A note on equimatchable graphs, Australas. J. Combin. 46 (2010) 185–190] has also provided a structural characterization for equimatchable graphs with girth at least five, in particular, a characterization for equimatchable bipartite graphs with girth at least six. In this paper, we extend the characterization of Frendrup by eliminating the girth condition. For an equimatchable graph, an edge is said to be a critical-edge if the graph obtained by the removal of this edge is not equimatchable. An equimatchable graph is called edge-critical, denoted by ECE, if every edge is critical. Noting that each ECE-graph can be obtained from some equimatchable graph by recursively removing non-critical edges, each equimatchable graph can also be constructed from some ECE-graph by joining some non-adjacent vertices. Our study reduces the characterization of equimatchable bipartite graphs to the characterization of bipartite ECE-graphs.

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等匹配二部图

如果图的所有最大匹配都具有相同的大小，则称图为等匹配图。Lesk等人【等匹配图，图论和组合学（Academic Press，London，1984）239–254】提供了等匹配二部图的特征。由于这种表征不是结构性的，Frendrup等人【关于等匹配图的注释，Australas.J.Combin.46（2010）185–190】还为周长至少为5的等匹配图提供了结构表征，特别是周长至少为6的等匹配二分图的表征。在本文中，我们通过消除周长条件来扩展Frendrup的特征。对于一个等匹配图，如果通过去除一条边得到的图是不可等匹配的，则称该边为临界边。如果每条边都是临界的，则等匹配图称为边临界图，用ECE表示。注意到每个ECE图都可以通过递归地去除非临界边从某个等匹配图中获得，每个等匹配图也可以通过连接一些不相邻的顶点从某个ECE图中构建。我们的研究将等匹配二部图的刻画简化为二部ECE图的刻画。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Discussiones Mathematicae Graph Theory MATHEMATICS-

CiteScore

2.20

自引率

0.00%

发文量

审稿时长

53 weeks

期刊介绍： The Discussiones Mathematicae Graph Theory publishes high-quality refereed original papers. Occasionally, very authoritative expository survey articles and notes of exceptional value can be published. The journal is mainly devoted to the following topics in Graph Theory: colourings, partitions (general colourings), hereditary properties, independence and domination, structures in graphs (sets, paths, cycles, etc.), local properties, products of graphs as well as graph algorithms related to these topics.