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Graph labellings and external difference families

Gavin Angus, Sophie Huczynska, Struan McCartney

TL;DR

This paper develops a systematic framework for using various types of vertex-labellings for graphs and digraphs to create digraph-defined external difference families.

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 $α$-valuations, namely near $α$-valuations and oriented near $α$-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 $α$-valuations for a family of trees without $α$-valuations, and an $α$-valuation for sun graphs).

Graph labellings and external difference families

TL;DR

This paper develops a systematic framework for using various types of vertex-labellings for graphs and digraphs to create digraph-defined external difference families.

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 -valuations, namely near -valuations and oriented near -valuations) with a graph blow-up technique. Many new families are produced, including the first explicit construction for an infinite family of -CEDFs, achieving all parameter sets for --CEDFs with sets. Further, new results arise for graph labellings themselves (e.g. cyclotomy-based near -valuations for a family of trees without -valuations, and an -valuation for sun graphs).
Paper Structure (14 sections, 34 theorems, 45 equations, 4 figures)

This paper contains 14 sections, 34 theorems, 45 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. $\beta$-valuations and digraph-defined EDFs
  4. Graph blow-up constructions
  5. Graceful labellings and corresponding EDFs
  6. $\alpha$-valuations of graphs
  7. Near $\alpha$-valuations of graphs
  8. Digraph-defined EDFs from near $\alpha$-valuations
  9. Oriented $\beta$- and near $\alpha$-valuations
  10. Digraph-defined EDFs with prescribed orientation
  11. $2$-CEDFs
  12. Ladder graphs
  13. Further Work
  14. Appendix: sun graphs

Key Result

Lemma 2.4

Let $\mathcal{A}=\{A_1, \ldots, A_m\}$ be a disjoint collection of $l$-subsets of a group $G$. Suppose $\mathcal{A}$ is a $c$-CEDF. Then, if $\gcd(c,m)=d$, the multiset in the definition may be written as a disjoint union of $d$ multiset unions, each involving $m/d$ sets, as follows (where all indic

Figures (4)

  • Figure 1: $S_{m,2}$
  • Figure 2: $B_{m,n}$
  • Figure 3: Example \ref{['nearalphaexmp']}
  • Figure 4: $S_{24}$, with a near $\alpha$-valuation.

Theorems & Definitions (90)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Definition 2.5
  • Lemma 2.6
  • Theorem 2.7
  • proof
  • Definition 2.8
  • Remark 2.9
  • ...and 80 more