Order plus size of $τ$-critical graphs
Let $G=(V,E)$ be a $τ$-critical graph with $τ(G)=t$. Erdős and Gallai proved that $|V|\leq 2t$ and the bound $|E|\leq {t+1\choose 2}$ was obtained by Erdős, Hajnal and Moon. We give here the sharp combined bound $|E|+|V|\leq {t+2\choose 2}$ and find all graphs with equality.
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