Maximum size of a triangle-free graph with bounded maximum degree and matching number

TL;DR

This paper determines the maximum edges in triangle-free graphs with bounded degree and matching number, showing restrictions cause decreases and proposing formulas and conjectures for various cases.

Contribution

It provides a formula for maximum edges in triangle-free graphs under degree and matching constraints, and introduces an integer programming approach for unresolved cases.

Findings

Forbidding triangles decreases maximum edges in extremal graphs.

A formula is given for cases where d ≥ m, and for d < m with specific bounds.

An integer programming formulation is proposed for remaining cases.

Abstract

Determining the maximum number of edges under degree and matching number constraints have been solved for general graphs by Chv\'{a}tal and Hanson (1976), and by Balachandran and Khare (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, respectively by Dibek, Ekim and Heggernes (2017), and by Blair, Heggernes, Lima and D.Lokshtanov (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 $d\geq m$, and for the cases where $d<m$ with either $d\leq 6$ or $Z(d)\leq m < 2d$ 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.