On diregular digraphs with degree two and excess three

TL;DR

This paper proves the non-existence of diregular digraphs with out-degree two and excess three for diameter at least three, advancing the classification of near-Moore digraphs with small excess.

Contribution

It establishes the first non-existence result for diregular digraphs with out-degree two and excess three, extending previous classifications for smaller excess.

Findings

No diregular digraphs with out-degree two and excess three exist for diameter k ≥ 3.

Completes the classification of digraphs close to Moore bounds for fixed out-degree.

Advances understanding of the structure and limitations of near-Moore digraphs.

Abstract

Moore digraphs, that is digraphs with out-degree $d$, diameter $k$ and order equal to the Moore bound $M(d,k) = 1 + d + d^2 + \dots +d^k$, arise in the study of optimal network topologies. In an attempt to find digraphs with a `Moore-like' structure, attention has recently been devoted to the study of small digraphs with minimum out-degree $d$ such that between any pair of vertices $u,v$ there is at most one directed path of length $\leq k$ from $u$ to $v$; such a digraph has order $M(d,k)+\epsilon $ for some small excess $\epsilon $. Sillasen et al. have shown that there are no digraphs with out-degree two and excess one. The present author has classified all digraphs with out-degree two and excess two. In this paper it is proven that there are no diregular digraphs with out-degree two and excess three for $k \geq 3$, thereby providing the first classification of digraphs with order three away from the Moore bound for a fixed out-degree.