Extremal triangle-free graphs with chromatic number at least four

TL;DR

The paper addresses extremal bounds for triangle-free graphs with chromatic number at least four. It develops a Mantel-type stability result and a vertex-deletion framework to reduce to bipartite cores, then proves a sharp bound $e(G)\le \left\lfloor\frac{(n-3)^2}{4}\right\rfloor+5$ for $n\ge 90$, with equality precisely for blow-ups of the Grötzsch graph. The main techniques combine stability analyses, a detailed case split on the matching number of a key subgraph, and a structural characterization leading to the family $\mathcal{G}(n)$ as the extremal constructions. This work sharpens classical results of Erdős–Gallai and Andrásfai, and identifies the extremal graphs as blow-ups of a specific 11-vertex triangle-free 4-chromatic graph.

Abstract

Let $G$ be an $n$-vertex triangle-free graph. The celebrated Mantel's theorem showed that $e(G)\leq \lfloor\frac{n^2}{4}\rfloor$. In 1962, Erdős (together with Gallai), and independently Andrásfai, proved that if $G$ is non-bipartite then $e(G)\leq \lfloor\frac{(n-1)^2}{4}\rfloor+1$. In this paper, we extend this result and show that if $G$ has chromatic number at least four and $n\geq 90$, then $e(G)\leq \lfloor\frac{(n-3)^2}{4}\rfloor+5$. The blow-ups of Grötzsch graph shows that this bound is best possible.

Figures (4)

• Figure 1: The Grötzsch graph $\Gamma$.
• Figure 2: The structure of graphs in $\mathcal{G}(n)$.
• Figure 3: The partition of $(A,B)$.
• Figure 4: The structure of $G$.

Theorems & Definitions (14)

• Theorem 1.1: Mantel
• Theorem 1.2: erdos2
• Theorem 1.3: brouwer
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1: Haggkvist
• Lemma 2.2
• proof
• Lemma 2.3
• proof