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A linear time algorithm for finding an Euler walk in a strongly connected 3-uniform hypergraph

Abstract : By an Euler walk in a 3-uniform hypergraph H we mean an alternating sequence v(0), epsilon(1), v(1), epsilon(2), v(2), ... , v(m-1), epsilon(m), v(m) of vertices and edges in H such that each edge of H appears in this sequence exactly once and v(i-1); v(i) is an element of epsilon(i), v(i-1) not equal v(i), for every i = 1, 2, ... , m. This concept is a natural extension of the graph theoretic notion of an Euler walk to the case of 3-uniform hypergraphs. We say that a 3-uniform hypergraph H is strongly connected if it has no isolated vertices and for each two edges e and f in H there is a sequence of edges starting with e and ending with f such that each two consecutive edges in this sequence have two vertices in common. In this paper we give an algorithm that constructs an Euler walk in a strongly connected 3-uniform hypergraph (it is known that such a walk in such a hypergraph always exists). The algorithm runs in time O(m), where m is the number of edges in the input hypergraph.
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Zbigniew Lonc, Pawel Naroski. A linear time algorithm for finding an Euler walk in a strongly connected 3-uniform hypergraph. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2012, Vol. 14 no. 1 (1), pp.147-158. ⟨hal-00990565⟩

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