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 $\tau_B(n)\leq n^2/25$.

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This page was last edited 14 October 2025.