k-uniform l-Eulerian hypergraphs
We consider $l$-Euler trails and $l$-Euler circuits in finite $k$-uniform directed multihypergraphs, where $k\geq 2$ and $1\leq l<k$. 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.