# Order plus size of $\tau$-critical graphs

## TL;DR

This paper establishes a sharp combined bound on the order and size of $ au$-critical graphs, extending previous bounds and characterizing all extremal graphs with equality.

## Contribution

It introduces a new combined bound on the number of vertices and edges in $ au$-critical graphs and characterizes all graphs that attain this bound.

## Key findings

- Proved the bound |E|+|V| ≤ (t+2 choose 2).
- Characterized all graphs with equality in the bound.
- Extended classical bounds by Erdős and Gallai, Hajnal, and Moon.

## Abstract

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