Consider the family of all finite graphs with maximum degree ∆(G) < d and matching number ν(G) < m. In this paper we give a new proof to obtain the exact upper bound for the number of edges in such graphs and also characterize all the cases when the maximal graph is unique. We also provide a new proof of Gallai’s lemma concerning factor critical graphs.

The matching preclusion number of a graph is the minimum number of neither edges whose deletion in a graph has a neither perfect matching nor an almost perfect matching. For many interconnection networks, the optimal sets are precisely those induced by a single vertex. Recently the conditional matching preclusion number of a graph was introduced to look for sets beyond those induced by a single...

We determine the 2-color Ramsey number of a connected triangle matching c(nK3) which is a graph with n vertex disjoint triangles plus (at least n − 1) further edges that keep these triangles connected. We obtain that R(c(nK3), c(nK3)) = 7n − 2, somewhat larger than in the classical result of Burr, Erdős and Spencer for a triangle matching, R(nK3, nK3) = 5n. The motivation is to determine the Ra...