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 hypergraphH 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 ε ei, vi-1 6 = 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. © 2012 Discrete Mathematics and Theoretical Computer Science (DMTCS).