Determining the maximum number of edges under degree and matching number constraints have been solved for general graphs in Chvátal and Hanson (J Combin Theory Ser B 20:128–138, 1976) and Balachandran and Khare (Discrete Math 309:4176–4180, 2009). It follows from the structure of those extremal graphs that deciding whether this maximum number decreases or not when restricted to claw-free graphs, to \(C_4\) -free graphs or to triangle-free graphs are separately interesting research questions. The first two cases being already settled in Dibek et al. (Discrete Math 340:927–934, 2017) and Blair et al. (Latin American symposium on theoretical informatics, 2020), in this paper we focus on triangle-free graphs. We show that unlike most cases for claw-free graphs and \(C_4\) -free graphs, forbidding triangles from extremal graphs causes a strict decrease in the number of edges and adds to the hardness of the problem. We provide a formula giving the maximum number of edges in a triangle-free graph with degree at most d and matching number at most m for all cases where \(d\ge m\) , and for the cases where \(d<m\) with either \(d\le 6\) or \(Z(d)\le m < 2d\) where Z(d) is a function of d which is roughly 5d/4. We also provide an integer programming formulation for the remaining cases and as a result of further discussion on this formulation, we conjecture that our formula giving the size of triangle-free extremal graphs is also valid for these open cases.
