The design spectrum of the Shrikhande graph

TL;DR

This paper determines the design spectrum of the Shrikhande graph and the line graph of K_{4,4}, showing they consist of all integers of the form 96t+1 for positive integers t.

Contribution

It proves that the design spectrum of these two specific graphs is exactly the set of integers 96t+1, expanding understanding of their combinatorial properties.

Findings

The design spectrum of the Shrikhande graph is {96t+1: t ≥ 1}.

The design spectrum of the line graph of K_{4,4} is {96t+1: t ≥ 1}.

Both spectra are characterized by a linear form 96t+1.

Abstract

The design spectrum of a simple graph $G$ is the set of positive integers $n$ such that there exists an edgewise decomposition of the complete graph $K_n$ into $n(n - 1)/(2 |E(G)|)$ copies of $G$. The purpose of this short paper is to prove that the Shrikhande graph and the line graph of $K_{4,4}$ have the design spectrum $\{96t + 1: t = 1, 2, \dots\}$.