Collapsible graphs and matchings

A graph G is collapsible if for every even subset R ⊆ V(G), there is a spanning connected subgraph of G whose set of odd degree vertices is R. A graph is reduced if it does not have nontrivial collapsible subgraphs. Collapsible and reduced graphs are defined and studied in [4]. In this article, we obtain a lower bound on the size of a maximum matching in a reduced graph. As an application, we verify and strengthen the Benhocine, Clark, Kohler, and Veldman conjecture [1], when restricted to 3-edge-connected graphs, by showing that for n large, a simple graph G with order n and with k′(G) ≥ 3 is collapsible or is contractible to the Petersen graph if for each edge uv ∈ E(G), d(u) + d(v) ≥ (n/5) − 2. We also characterize the extremal graphs. © 1993 John Wiley & Sons, Inc.