k-uniform l-Eulerian hypergraphs
Abstract
We consider l-Euler trails and l-Euler circuits in finite k-uniform directed multihypergraphs, where k≥ 2 and 1≤ l<k. Let D be such a directed multihypergraph. The notions of l-connectivity and l-balance are defined in terms of the l-tails and l-heads of its hyperarcs. It is shown that D is l-Eulerian if and only if D is l-connected and l-balanced. Moreover, D is l-semi-Eulerian if and only if D is l-connected and has exactly two states with unequal l-indegrees and l-outdegrees, one with l-outdegree one greater than its l-indegree and the other with l-indegree one greater than its l-outdegree. The proofs are based on an l-transition digraph and the classical Euler theorem for directed multigraphs. The corresponding conditions for k-uniform multihypergraphs are also obtained in terms of their orientations. For a given orientation, an expected linear-time algorithm is provided to determine whether an l-Euler trail or an l-Euler circuit exists and to construct one when it does.
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BibTeX
@misc{lei2026kuniform,
title = {{k-uniform l-Eulerian hypergraphs}},
author = {Yiming Lei and Chenxi Lv and Bing Dai},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02742},
eprint = {2611.02742},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02742},
}
RIS
TY - UNPB
TI - k-uniform l-Eulerian hypergraphs
AU - Yiming Lei
AU - Chenxi Lv
AU - Bing Dai
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02742
UR - https://arxiv.org/abs/2611.02742
AB - We consider l-Euler trails and l-Euler circuits in finite k-uniform directed multihypergraphs, where k≥ 2 and 1≤ l<k. Let D be such a directed multihypergraph. The notions of l-connectivity and l-balance are defined in terms of the l-tails and l-heads of its hyperarcs. It is shown that D is l-Eulerian if and only if D is l-connected and l-balanced. Moreover, D is l-semi-Eulerian if and only if D is l-connected and has exactly two states with unequal l-indegrees and l-outdegrees, one with l-outdegree one greater than its l-indegree and the other with l-indegree one greater than its l-outdegree. The proofs are based on an l-transition digraph and the classical Euler theorem for directed multigraphs. The corresponding conditions for k-uniform multihypergraphs are also obtained in terms of their orientations. For a given orientation, an expected linear-time algorithm is provided to determine whether an l-Euler trail or an l-Euler circuit exists and to construct one when it does.
ER -