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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2603.05662 |
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| _version_ | 1866910042883620864 |
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| author | Angus, Gavin Huczynska, Sophie McCartney, Struan |
| author_facet | Angus, Gavin Huczynska, Sophie McCartney, Struan |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_05662 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Graph labellings and external difference families Angus, Gavin Huczynska, Sophie McCartney, Struan Combinatorics 05B10 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). |
| title | Graph labellings and external difference families |
| topic | Combinatorics 05B10 |
| url | https://arxiv.org/abs/2603.05662 |