Anti-Eulerian digraphs

Abstract

Let D be a digraph. A closed anti-directed Euler trail of D is a closed trail in which consecutive arcs have opposite directions and each arc of D occurs exactly once. A digraph is anti-Eulerian if it contains a closed anti-directed Euler trail. A tournament T is anti-Eulerian if and only if both d+(v) and d−(v) are even for any v∈V(T). The Cartesian product of two directed cycles Cn1→□Cn2→ is anti-Eulerian if and only if gcd(n1,n2)=1. If n1,n2,…,n2k can be partitioned into k relatively prime pairs, then Cn1→□Cn2→□⋯□Cn2k→ is anti-Eulerian. If each of Di (1≤i≤k) is an anti-Eulerian digraph and at least one of them is not a bipartite digraph, then D1□D2□⋯□Dk is anti-Eulerian.