Maximum size of a trianglefree graph with bounded maximum degree and matching number
Abstract
Determining the maximum number of edges under degree and matching number constraints have been solved for general graphs by Chvátal 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 clawfree graphs, to $C_4$free graphs or to trianglefree 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 trianglefree graphs. We show that unlike most cases for clawfree 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 trianglefree 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 trianglefree extremal graphs is also valid for these open cases.
 Publication:

arXiv eprints
 Pub Date:
 July 2022
 arXiv:
 arXiv:2207.02271
 Bibcode:
 2022arXiv220702271A
 Keywords:

 Mathematics  Combinatorics;
 05C35;
 05C55
 EPrint:
 22 pages, 3 figures