Constructing extremal triangle-free graphs using integer programming

Abstract

The maximum number of edges in a graph with matching number $m$ and maximum degree $d$ has been determined in Chvátal and Hanson (1976) and Balachandran and Khare (2009), where some extremal graphs have also been provided. Then, a new question has emerged: how the maximum edge count is affected by forbidding some subgraphs occurring in these extremal graphs? In Ahanjideh et al. (2022), the problem is solved in triangle-free graphs for $d\ge m$, and for $d<m$ with either $Z\left(d\right)\le m<2d$ or $d\le 6$, where $Z\left(d\right)$ is approximately $5d/4$. The authors derived structural properties of triangle-free extremal graphs, which allows us to focus on constructing small extremal components to form an extremal graph. Based on these findings, in this paper, we develop an integer programming formulation for constructing extremal graphs. Since our formulation is highly symmetric, we use our own implementation of Orbital Branching to reduce symmetry. We also implement our integer programming formulation so that the feasible region is restricted iteratively. Using a combination of the two approaches, we expand the solution into $d\le 10$ instead of $d\le 6$ for $m>d$. Our results endorse the formula for the number of edges in all extremal triangle-free graphs conjectured in Ahanjideh et al. (2022).