Contractible Edges in k-Connected Infinite Graphs

Abstract

In this paper, we prove that every vertex in a k-connected locally finite graph $(k\ge 2)$ which is triangle-free or has minimum degree greater than $\frac{3}{2}(k-1)$ is incident to at least two contractible edges. Also, it is shown that every vertex in a k-connected locally finite graph $(k\ge 3)$ with no adjacent triangles is incident to a contractible edge. By restricting to graphs with large minimum end vertex-degree, we generalize Egawa’s result (Graphs Comb 7:15–21, 1991) and prove that every k-connected locally finite infinite graph such that the minimum degree is at least $\lfloor \frac{5k}{4}\rfloor$ and all ends have vertex-degree greater than k contains a contractible edge. We also generalize Dean’s result (J Comb Theory Ser B 48:1–5, 1990) and prove that for any k-connected locally finite infinite graph G $(k\ge 4)$ with minimum end vertex-degree greater than k which is triangle-free or has minimum degree at least $\lfloor \frac{3k}{2}\rfloor$, the closure of the subgraph induced by all the contractible edges in the Freudenthal compactification of G is topologically 2-connected.