Extremal Aspects of the Erd\H{o}s--Gallai--Tuza Conjecture

TL;DR

This paper investigates extremal properties related to the Erd ext{H}o}s-Gallai-Tuza conjecture, establishing structural conditions for potential counterexamples and proving the conjecture for specific classes of graphs.

Contribution

It proves that minimal counterexamples must have high minimum degree and dense edge cuts, and confirms the conjecture for graphs without an induced $K_4^-$ subgraph.

Findings

Minimal counterexamples have minimum degree > n/2

Conjecture holds for graphs with no induced $K_4^-$

Conjecture reduces to triangular graphs if true generally

Abstract

Erd\H{o}s, Gallai, and Tuza posed the following problem: given an $n$-vertex 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 a set of edges containing at most one edge from each triangle of $G$. Is it always the case that $\alpha_1(G) + \tau_1(G) \leq n^2/4$? We also consider a variant on this conjecture: if $\tau_B(G)$ is the smallest size of an edge set whose deletion makes $G$ bipartite, does the stronger inequality $\alpha_1(G) + \tau_B(G) \leq n^2/4$ always hold? By considering the structure of a minimal counterexample to each version of the conjecture, we obtain two main results. Our first result states that any minimum counterexample to the original Erd\H{o}s--Gallai--Tuza Conjecture has "dense edge cuts", and in particular has minimum degree greater than $n/2$. This implies that the conjecture holds for all graphs if and only if it holds for all triangular graphs (graphs where every edge lies in a triangle). Our second result states that $\alpha_1(G) + \tau_B(G) \leq n^2/4$ whenever $G$ has no induced subgraph isomorphic to $K_4^-$, the graph obtained from the complete graph $K_4$ by deleting an edge. Thus, the original conjecture also holds for such graphs.