A construction for regular-graph designs

TL;DR

This work advances regular-graph designs by presenting a foundational construction for λ=1 designs on a δ-regular graph with block size k=δ+1, leveraging neighbourhood blocks and remainder blocks under girth constraints to control edge and non-edge pair multiplicities. It integrates group-divisible designs and Wilson’s fundamental construction to build designs for broad families of orders, providing explicit results for δ∈{2,3} and constructing connected 4-regular designs at orders 105 and 117. The paper also delivers extensive computational nonexistence results for higher-regularity cases and delineates the practical limits of the basic construction, highlighting open cases (notably certain small orders and connectedness questions). Overall, it bridges design theory techniques (GDDs, Wilson constructions) with regular graphs to yield concrete design orders and explicit block lists, advancing the understanding of how graph structure governs block-design realizability.

Abstract

A regular-graph design is a block design for which a pair $\{a,b\}$ of distinct points occurs in $λ+1$ or $λ$ blocks depending on whether $\{a,b\}$ is or is not an edge of a given $δ$-regular graph. Our paper describes a specific construction for regular-graph designs with $λ= 1$ and block size $δ+ 1$. We show that for $δ\in \{2,3\}$, certain necessary conditions for the existence of such a design with $n$ points are sufficient, with two exceptions in each case and two possible exceptions when $δ= 3$. We also construct designs of orders 105 and 117 for connected 4-regular graphs.

Figures (1)

• Figure 1: Pair occurrence arrays for $C_9$, $C_{11}$ and $C_5 + C_6$

Theorems & Definitions (16)

• Lemma 2.1
• proof
• Lemma 3.1
• proof
• Theorem 4.1
• proof
• Theorem 4.2
• proof
• Lemma 4.1
• proof