Matchings in Graphs of Odd Regularity and Girth

Petersen proved that every 3-regular brigdeless graph has a perfect matching. Motivated by this result, several authors established lower bounds on the matching number of a graph subject to degree and girth conditions. In 2012 Henning et al. proved such a lower bound for connected 3-regular graphs. In the present paper, we extend their result by establishing lower bounds on the matching number of graphs of given odd regularity d and odd girth g, which are sharp for many values of d and g. For d = g = 5 , we characterize all extremal graphs.