Some results on generalized strong external difference families

TL;DR

This paper introduces new generalized strong external difference families (GSEDFs), explores their applications in graph decomposition, and establishes nonexistence results for certain parameter sets, advancing combinatorial design theory.

Contribution

The paper constructs new GSEDFs for m=2, applies them to graph decomposition, and proves nonexistence of specific GSEDFs when sum of parameters is less than v.

Findings

New GSEDF constructions for m=2

Application of GSEDFs to graph decomposition

Nonexistence of certain GSEDFs when sum of k's is less than v

Abstract

A generalized strong external difference family (briefly $(v, m; k_1,\dots,k_m; \lambda_1,\dots,\lambda_m)$-GSEDF) was introduced by Paterson and Stinson in 2016. In this paper, we construct some new GSEDFs for $m=2$ and use them to obtain some results on graph decomposition. We also give some nonexistence results for GSEDFs. Especially, we prove that a $(v, 3;k_1,k_2,k_3; \lambda_1,\lambda_2,\lambda_3)$-GSEDF does not exist when $k_1+k_2+k_3< v$.