Graph labellings and external difference families

TL;DR

This paper introduces a systematic framework using graph vertex-labellings to construct external difference families, leading to new explicit constructions and insights into graph labellings and their applications in combinatorics and cryptography.

Contribution

It develops a unified approach combining vertex-labellings and graph blow-up techniques to generate new external difference families, including the first explicit infinite family of 2-CEDFs.

Findings

Constructed the first explicit infinite family of 2-CEDFs with certain parameters.

Developed new graph labellings, including cyclotomy-based near α-valuations.

Produced new results for specific graph classes like trees and sun graphs.

Abstract

Digraph-defined external difference families were recently introduced as a natural generalization of several well-studied combinatorial objects motivated by cryptography (e.g. external difference families (EDFs) and circular external difference families (CEDFs)). In this paper, we develop a systematic framework for using various types of vertex-labellings for graphs and digraphs to create digraph-defined external difference families. The approach is to combine suitable vertex-labellings (generalizations of $\alpha$-valuations, namely near $\alpha$-valuations and oriented near $\alpha$-valuations) with a graph blow-up technique. Many new families are produced, including the first explicit construction for an infinite family of $2$-CEDFs, achieving all parameter sets for $(n,m,l;1)$-$2$-CEDFs with $m \equiv 0 \mod 4$ sets. Further, new results arise for graph labellings themselves (e.g. cyclotomy-based near $\alpha$-valuations for a family of trees without $\alpha$-valuations, and an $\alpha$-valuation for sun graphs).