Honors Theses

Title

Edge Idependence in Graphs

Summer 4-13-1995

Department

Mathematics

Gary Chartrand, Mathematics and Statistics

Michelle Schultz, Mathematics and Statistics

Dawn Jones, Mathematics and Statistics

Abstract

Two edges in a graph are independent if they are not adjacent. A set of edges is independent if every two edges in the set are independent. A set of independent edges in a graph G is also called a matching in G. The greatest number of independent edges in a graph G is denoted by β1(G). A matching of cardinality β1(G) is called a maximum matching in G. The number βˉ1(G) is the minimum cardinality among the maximal matchings of G. Bounds are established for β1(G) in terms of βˉ1(G); namely, for every nonempty graph G, βˉ1(G)≤β1(G)≤2βˉ1(G). Furthermore, for every two positive integers a and b with ab≤2a, there exists a graph G of order 2b such that βˉ1(G)=a and β1(G)=b. Moreover, the graph G with this property can be required to be connected or even 2-connected. It is also shown that for every graph G and every integer k with βˉ1(Gk˂β1(G), there exists a maximal matching in G with k edges. Sharp upper and lower bounds are presented for βˉ1(G)+βˉ1(Gˉ). The maximum matchings of a graph G can themselves be represented by a graph. The matching graph M(G) of a graph is that graph whose vertices are the maximum matchings of G and two vertices M0 and M1 are adjacent in M(G) if M0 and M1 differ by one edge. A graph H is a matching graph if there is a graph G such that H=M(G). The problem of determining which graphs are matching graphs is studied. Several graphs are shown to be matching graphs, and it is also shown that not all graphs are matching graphs.

Access Setting

Honors Thesis-Campus Only

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