An invariant for minimum triangle-free graphs

TL;DR

This paper introduces a new invariant involving edges, vertices, independence number, and 4-cycle counts in triangle-free graphs, establishing a key inequality and characterizing extremal cases.

Contribution

It presents a novel inequality involving multiple graph parameters for triangle-free graphs and characterizes the graphs achieving equality.

Findings

Proves the inequality 3e(G) - 17n(G) + 35α(G) + N(C4;G) ≥ 0 for all triangle-free graphs.

Characterizes the graphs where equality holds.

Provides insights into the structure of extremal triangle-free graphs.

Abstract

We study the number of edges, $e(G)$, in triangle-free graphs with a prescribed number of vertices, $n(G)$, independence number, $\alpha(G)$, and number of cycles of length four, $\operatorname{N}(C_4;G)$. We in particular show that $$3e(G) - 17n(G) + 35\alpha(G) + \operatorname{N}(C_4;G) \geq 0$$ for all triangle-free graphs $G$. We also characterise the graphs that satisfy this inequality with equality.