Discrete Mathematics & Theoretical Computer Science, Vol 14, No 1 (2012)

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

Zbigniew Lonc, Paweł Naroski


By an Euler walk in a 3-uniform hypergraph H we mean an alternating sequence v0,e1,v1,e2,v2,…,vm-1,em,vm of vertices and edges in H such that each edge of H appears in this sequence exactly once and vi-1,vi∈ei, vi-1¬=vi , 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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