The structure of typical eye-free graphs and a Turan-type result for two weighted colours

TL;DR

This paper characterizes the structure of typical graphs avoiding a specific induced subgraph called the $(a,b)$-eye, showing they are close to certain multipartite graphs, and establishes a Turán-type extremal result for a weighted coloring problem.

Contribution

It introduces a stability framework for $(a,b)$-eye-free graphs in the Erdős–Rényi model and proves a new Turán-type extremal theorem for a two-colored weighted graph problem.

Findings

Typical $(a,b)$-eye-free graphs are close to $a$-partite or complement of $(b-1)$-partite graphs.

The use of hypergraph containers effectively characterizes the structure of these graphs.

An exact Turán-type extremal result for the weighted coloring problem is established.

Abstract

The $(a,b)$-eye is the graph $I_{a,b} = K_{a+b}-K_b$ obtained by deleting the edges of a clique of size $b$ from a clique of size $a+b$. We show that for any $a,b \ge 2$ and $p \in (0,1)$, if we condition the random graph $G \sim G(n,p)$ on having no induced copy of $I_{a,b}$, then with high probability $G$ is close to an $a$-partite graph or the complement of a $(b-1)$-partite graph. Our proof uses the recently developed theory of hypergraph containers, and a stability result for an extremal problem with two weighted colours. We also apply the stability method to obtain an exact Tur\'an-type result for this extremal problem.