A characterization of graphs with given maximum degree and smallest possible matching number: II

Abstract

Let α′(G) be the matching number of a graph G. A characterization of the graphs with given maximum odd degree and smallest possible matching number is given by Henning and Shozi (2021) [13]. In this paper we complete our study by giving a characterization of the graphs with given maximum even degree and smallest possible matching number. In 2018 Henning and Yeo [10] proved that if G is a connected graph of order n, size m and maximum degree k where k≥4 is even, then α′(G)≥nk(k+1)+mk+1−1k(k+1), unless G is k-regular and n∈{k+1,k+3}. In this paper, we give a complete characterization of the graphs that achieve equality in this bound when the maximum degree k is even, thereby completing our study of graphs with given maximum degree and smallest possible matching number.