Graph labellings and 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).

• The paper shows that near α-valuations and oriented near α-valuations convert graph labellings into EDFs through balanced blow-ups, producing (|E|l²+1, |V|, l, 1)-EDFs.
• The paper constructs near α-valuations beyond the α-valuation class and proves that every prime p ≥ 11 with p ≠ ±1 mod 8 yields a tree with a near α-valuation but no α-valuation.
• The paper gives the first explicit infinite family of 2-CEDFs with parameters (4kl²+1, 4k, l, 1) for all k,l ≥ 1, resolving the m ≡ 0 mod 4 case while leaving m ≡ 2 mod 4 open.

Overview

This paper develops a systematic connection between vertex-labellings of graphs and digraph-defined external difference families (EDFs), a framework introduced by Huczynska, Jefferson, and McCartney that subsumes classical EDFs, strong EDFs, and circular external difference families (CEDFs) by specifying difference conditions via a labelled digraph (2603.05662). The central observation is that a $\beta$-valuation (graceful labelling) of an $n$-edge graph $G$ immediately yields an $(n,m,1,1;G^b)$-EDF in $\mathbb{Z}_{n+1}$: orienting each edge toward the larger label (the "natural orientation") converts the multiset of absolute edge differences $\{1,\ldots,n\}$ into directed differences covering every non-zero group element exactly once. To obtain EDFs with block size $l>1$, however, the bare graceful condition is insufficient; the paper identifies the necessary extra structure and generalizes it.

Near $\alpha$-valuations as the key condition

The required strengthening is the near $\alpha$-valuation: a $\beta$-valuation whose vertices partition into $n$0 and $n$1 such that each vertex's label is respectively smaller (or larger) than all its neighbours' labels. Under the natural orientation this is precisely a source-sink orientation, and it forces bipartiteness. Every $n$2-valuation is a near $n$3-valuation, but not conversely: Rosa's 7-vertex tree $n$4 admits a near $n$5-valuation while lacking any $n$6-valuation. This distinction matters because the class of graphs with near $n$7-valuations strictly extends those with $n$8-valuations, and the open conjecture that every tree has a near $n$9-valuation (verified computationally up to 20 vertices) would, if resolved affirmatively, guarantee $G$0-EDFs for all trees via the machinery below.

The paper contributes new labelling results in this direction. A cyclotomy-based construction produces, for every prime $G$1 with $G$2, a tree possessing a near $G$3-valuation but no $G$4-valuation — these trees lie in Rosa's obstructed class $G$5, so the obstruction to $G$6-valuations is genuine rather than an artifact of the construction. Additionally, an appendix gives what the authors believe to be a new $G$7-valuation for sun graphs $G$8, with threshold value $G$9 separating the bipartition classes.

Blow-up constructions for larger block sizes

The core engine is a blow-up argument. Given a graph or digraph with a (near) $(n,m,1,1;G^b)$0-type valuation, replacing each vertex by a set of $(n,m,1,1;G^b)$1 vertices — equivalently taking the lexicographic product with the empty graph $(n,m,1,1;G^b)$2 — preserves the valuation type, with edge differences filling contiguous intervals $(n,m,1,1;G^b)$3 that tile $(n,m,1,1;G^b)$4. The proofs split into blowing up $(n,m,1,1;G^b)$5 versus $(n,m,1,1;G^b)$6 separately, exploiting the fact that the valuation guarantees label ordering across the bipartition. A key lemma shows the blow-up respects the natural orientation: if $(n,m,1,1;G^b)$7, all labels in the cluster $(n,m,1,1;G^b)$8 are smaller than all labels in $(n,m,1,1;G^b)$9, so orientations are inherited consistently.

The resulting main theorem states: if $\mathbb{Z}_{n+1}$0 has a near $\mathbb{Z}_{n+1}$1-valuation, there exists an $\mathbb{Z}_{n+1}$2-EDF in $\mathbb{Z}_{n+1}$3. An explicit example with a 5-edge graph blown up by $\mathbb{Z}_{n+1}$4 yields a $\mathbb{Z}_{n+1}$5-EDF in $\mathbb{Z}_{n+1}$6. All constructions achieve $\mathbb{Z}_{n+1}$7; doubling the edge set with its reverse yields $\mathbb{Z}_{n+1}$8 trivially.

Oriented valuations

Where no suitable undirected labelling exists, the paper employs oriented $\mathbb{Z}_{n+1}$9-valuations (Bloom–Hsu graceful digraphs): injective vertex maps whose directed modular edge differences cover $\{1,\ldots,n\}$0. These can exist even when the underlying graph has no $\{1,\ldots,n\}$1-valuation — notably for cycles $\{1,\ldots,n\}$2 with $\{1,\ldots,n\}$3, where reversing one edge of the natural orientation repairs the single duplicated/missing label pair $\{1,\ldots,n\}$4. A general "flip" lemma shows that reversing edges whose labels form pairs $\{1,\ldots,n\}$5 preserves the property, and scaling/translation lemmas generate further valuations. A second cyclotomic construction gives oriented $\{1,\ldots,n\}$6-valuations for stars $\{1,\ldots,n\}$7 using primitive elements of $\{1,\ldots,n\}$8, with the case split determined by whether $\{1,\ldots,n\}$9 is a square.

Introducing oriented near $l>1$0-valuations (the oriented analogue with the source-sink bipartition condition), the blow-up theorems extend verbatim: any digraph with such a valuation yields an $l>1$1-EDF. Here the proof must handle negative integer differences arising from wraparound modulo $l>1$2, tracking them through the blow-up before reducing mod $l>1$3.

Prescribed orientations and 2-CEDFs

Since CEDFs require a unidirectional clockwise cycle orientation, the final section adapts known $l>1$4-valuations to fixed target orientations. For unidirectional paths, ladders (left-to-right, bottom-to-top), and semi-directed sun digraphs, explicit oriented near $l>1$5-valuations are obtained from existing $l>1$6-valuations via the flip lemma, yielding corresponding infinite EDF families.

The headline result concerns 2-CEDFs. Prior work contained no direct explicit construction of infinite families of 2-CEDFs (only a cyclotomic construction of Wu, Yang, and Feng with parameter-dependent $l>1$7). Using Abrham–Kotzig $l>1$8-valuations for $l>1$9 ($\alpha$0, plus small cases $\alpha$1) and Kotzig $\alpha$2-valuations for $\alpha$3 ($\alpha$4, plus $\alpha$5), the authors verify in each case that flipping the appropriate edge-label classes converts the natural orientation into the clockwise orientation while preserving the full difference coverage. Applying the blow-up theorem and interleaving the two cycles' blocks gives:

$\alpha$6

This covers all parameter sets $\alpha$7-2-CEDFs with $\alpha$8. Since a 2-CEDF with odd $\alpha$9 is a re-indexed 1-CEDF, only the case $\alpha$0 remains open within this framework. An explicit $\alpha$1-2-CEDF in $\alpha$2 illustrates the construction end-to-end.

Limitations and open questions

Several constraints bound the method's scope. The source-sink condition forces bipartiteness, so the valuation-then-blow-up pipeline cannot directly handle non-bipartite graphs despite their possible $\alpha$3-valuations; whether an analogous method exists for them is posed as an open question. Whether every bipartite graph with a $\alpha$4-valuation admits some near $\alpha$5-valuation remains unresolved (counterexamples exist for specific labellings but not for the graphs themselves). The weak tensor product closure raises the question of whether products of two non-$\alpha$6 graphs remain non-$\alpha$7. Finally, all constructed families have $\alpha$8; extending the technique to other $\alpha$9 values is left open. The orientation-standardization approach also fails in some cases — square lattice graphs and symmetric trees carry $\beta$0-valuations whose $\beta$1-labelled edge pairs cannot be simultaneously flipped to the desired orientation.

Conclusion

The paper establishes that near $\beta$2-valuations (and their oriented analogues) are exactly the additional structure beyond gracefulness needed to convert vertex-labellings into digraph-defined EDFs via balanced blow-ups, and demonstrates the framework's reach with the first explicit infinite family of 2-CEDFs, resolving all cases with $\beta$3. The remaining gap at $\beta$4, together with the bipartiteness restriction and the $\beta$5 limitation, delineates the precise boundary of the current technique.