A generalization of Petersen's matching theorem

Abstract

One of the earliest results in graph theory is Petersen's matching theorem from 1891 which states that every bridgeless cubic graph contains a perfect matching. Since the vertex-connectivity and edge-connectivity in a connected cubic graph are equal, Petersen's theorem can be stated as follows: If G is a 2-connected 3-regular graph of order n, then α′(G)=12n, where α′(G) denotes the matching number of G. We generalize Petersen's theorem and prove that for k≥3 an odd integer, if G is a 2-connected k-regular graph of order n, then α′(G)≥(k2+k+6k2+2k+3)×n2. The case when k is even behaves differently. In this case, for k≥4 even, if G is a 2-connected k-regular graph of order n, then α′(G)≥(k2+4k2+k+2)×n2. For all k≥3, if G is a 2-connected graph of order n and maximum degree k that is not necessarily regular, then we show that α′(G)≥2nk+2. In all the above bounds, the extremal graphs are characterized.