# Graphs with $\alpha_1$ and $\tau_1$ both large

## TL;DR

This paper explores the relationship between two graph parameters,  and , showing that both can be large simultaneously in certain graphs, challenging previous conjectures about their combined bounds.

## Contribution

The paper investigates three problems related to  and , providing constructions of graphs where these parameters are both large, thus extending understanding of their interplay.

## Key findings

- Constructed graphs with large  and  values
- Demonstrated limitations of existing upper bounds
- Extended the theoretical framework of graph parameter relationships

## Abstract

Given a graph $G$, let $\tau_1(G)$ denote the smallest size of a set of edges whose deletion makes $G$ triangle-free, and let $\alpha_1(G)$ denote the largest size of an edge set containing at most one edge from each triangle of $G$. Erd\H{o}s, Gallai, and Tuza introduced several problems with the unifying theme that $\alpha_1(G)$ and $\tau_1(G)$ cannot both be "very large"; the most well-known such problem is their conjecture that $\alpha_1(G) + \tau_1(G) \leq |V(G)|^2/4$, which was proved by Norin and Sun. We consider three other problems within this theme (two introduced by Erd\H{o}s, Gallai, and Tuza, another by Norin and Sun), all of which request an upper bound either on $\min\{\alpha_1(G), \tau_1(G)\}$ or on $\alpha_1(G) + k\tau_1(G)$ for some constant $k$, and prove the existence of graphs for which these quantities are "large".