[论文解读] Graph labellings and external difference families

Gavin Angus, Sophie Huczynska+1 more|arXiv (Cornell University)|Mar 5, 2026

graph theory and CDMA systems被引用 0

一句话总结

该论文构建了一个框架，使用图形与有向图的顶点标标记（特别是接近 α-值与有向近 α-值）结合图的吹大来构造以有向图定义的外部差分族（EDF），并带来包括 2-CEDFs 在内的新的无限族。

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).

研究动机与目标

• 通过图标记技术为外部差分族（EDF）提供动机与推广。
• develop a systematic method to turn vertex-labellings of graphs/digraphs into digraph-defined EDFs.
• Exploit graph blow-up operations to extend known constructions and produce new families of EDFs and CEDFs.
• Present first explicit infinite family of 2-CEDFs covering all parameter sets with m ≡ 0 (mod 4).
• Explore new valuation results (near α-valuations) and their impact on EDF constructions.

提出的方法

• Use β-valuations and near α-valuations (and oriented variants) to induce digraphs with prescribed difference multisets.
• Orient edges according to label order (natural orientation) to realize EDF conditions.
• Apply graph blow-ups and lexicographic products to extend labellings to larger graphs while preserving EDF properties.
• Leverage cyclotomy-based constructions to obtain near α-valuations for trees lacking α-valuations.
• Derive EDFs in cyclic groups Z_{n+1} from labeled graphs via Δ(A_j,A_i) multisets.
• Provide proofs showing how near α-valuations yield (n,m,l,λ;G^p)-EDF structures in Z_{n+1}.

实验结果

研究问题

• RQ1 How can graph and digraph vertex-labellings be used to systematically construct digraph-defined EDFs?
• RQ2 What labelling properties are essential to ensure the required difference multisets cover each nonzero group element exactly once?
• RQ3 Can blow-up and product operations preserve near α-valuations and thereby extend EDF constructions to larger families?
• RQ4 What new infinite families of CEDFs/2-CEDFs can be obtained from near α-valuations and oriented labellings?
• RQ5 Do cyclotomy-based labellings yield trees with near α-valuations that expand the known EDF repertoire?

主要发现

• Established a framework linking graph-labelled digraphs to digraph-defined EDFs in cyclic groups.
• Proved that a graph with a near α-valuation yields an (|E|l^2+1,|V|,l,1;G^p)-EDF in Z_{|E|l^2+1} through blow-up constructions.
• Provided the first explicit construction of infinite families of 2-CEDFs covering all parameter sets with m ≡ 0 (mod 4).
• Demonstrated that near α-valuations can exist on graphs where α-valuations do not, including constructions for certain trees and cycles.
• Showed that lexicographic and weak tensor product operations preserve near α-valuations, enabling broader EDF generation.
• Offered cyclotomy-based near α-valuations yielding trees without α-valuations but with near α-valuations.