Disproving the normal graph conjecture

TL;DR

This paper disproves a longstanding conjecture by showing that certain graphs without specific induced subgraphs are not necessarily normal, challenging previous assumptions in graph theory.

Contribution

The paper provides a counterexample to the 1999 conjecture that graphs excluding certain cycles are always normal.

Findings

Disproved the 1999 conjecture on graph normality.

Identified graphs without $C_5$, $C_7$, and $ar{C}_7$ that are not normal.

Challenged previous beliefs about the structure of normal graphs.

Abstract

A graph $G$ is called normal if there exist two coverings, $\mathbb{C}$ and $\mathbb{S}$ of its vertex set such that every member of $\mathbb{C}$ induces a clique in $G$, every member of $\mathbb{S}$ induces an independent set in $G$ and $C \cap S \neq \emptyset$ for every $C \in \mathbb{C}$ and $S \in \mathbb{S}$. It has been conjectured by De Simone and K\"orner in 1999 that a graph $G$ is normal if $G$ does not contain $C_5$, $C_7$ and $\overline{C_7}$ as an induced subgraph. We disprove this conjecture.