On Regular \((2,q)\)-Extendable Graphs

Abstract

Let \(G\) be a graph with a maximum matching of size \(q\), and let \(p \leq q\) be a positive integer. Then \(G\) is called \((p, q)\)-extendable if every set of \(p\) independent edges can be extended to a matching of size \(q\). If \(G\) is a graph of even order \(n\) and \(n = 2q\), then \((p,q)\)-extendable graphs are exactly the \(p\)-extendable graphs defined by Plummer \([11]\) in \(1980\).

Let \(d \geq 3\) be an integer, and let \(G\) be a \(d\)-regular graph of order \(n\) with a maximum matching of size \(q = \frac{n-t}{2}\geq 3\) for an integer \(t \geq 1\) such that \(n – t\) is even. In this work, we prove that if

(i) \(n \leq {(t+4)(d+1)-5}\) or

(ii) \(n \leq (t+4)(d+2) – 1\) when \(d\) is odd,

then \(G\) is \((2, q)\)-extendable.