Almost All Regular Graphs are Normal

TL;DR

This paper proves that all graphs with bounded maximum degree and large enough odd girth are normal, confirming the Normal Graph Conjecture for a broad class of graphs and showing that random regular graphs are almost surely normal.

Contribution

It establishes that graphs with bounded degree and large odd girth are normal, advancing the understanding of the Normal Graph Conjecture.

Findings

Graphs with bounded degree and large odd girth are normal.

Random d-regular graphs are almost surely normal.

Supports the Normal Graph Conjecture for specific graph classes.

Abstract

In 1999, De Simone and K\"{o}rner conjectured that every graph without induced $C_5,C_7,\overline{C}_7$ contains a clique cover $\mathcal C$ and a stable set cover $\mathcal I$ such that every clique in $\mathcal C$ and every stable set in $\mathcal I$ have a vertex in common. This conjecture has roots in information theory and became known as the Normal Graph Conjecture. Here we prove that all graphs of bounded maximum degree and sufficiently large odd girth (linear in the maximum degree) are normal. This implies that for every fixed $d$, random $d$-regular graphs are a.a.s. normal.