Problem Statement
Let $G$ be a graph on $n$ vertices, $\alpha_1(G)$ be the maximum number of edges that contain at most one edge from every triangle, and $\tau_1(G)$ be the minimum number of edges that contain at least one edge from every triangle.
Is it true that\[\alpha_1(G)+\tau_1(G) \leq \frac{n^2}{4}?\]
Is it true that\[\alpha_1(G)+\tau_1(G) \leq \frac{n^2}{4}?\]
Categories:
Graph Theory
Progress
A problem of Erdős, Gallai, and Tuza [EGT96], who observe that this is probably quite difficult since there are different examples where equality hold: the complete graph, the complete bipartite graph, and the graph obtained from $K_{m,m}$ by adding one vertex joined to every other.This is true, and was proved by Norin and Sun [NoSu16], who in fact proved that\[\alpha_1(G)+\tau_B(G) \leq \frac{n^2}{4},\]where $\tau_B(G)$ is the minimum number of edges that need to be removed to make the graph bipartite. (Note that clearly $\tau_1(G)\leq \tau_B(G)$.) Problem [23] can be phrased as the conjecture $\tau_B(n)\leq n^2/25$.
Source: erdosproblems.com/621 | Last verified: January 15, 2026