Necessary and Sufficient Conditions for Circulant Digraphs to be Antistrong, Weakly-Antistrong and Anti-Eulerian

Abstract

An antidirected trail in a digraph is a trail (a walk with no arc repeated) in which the consecutive arcs have opposite directions. The forward antidirected trails are special antidirected trails which begin and end with forward arcs. A digraph D containing a forward antidirected (x, y)-trail for every choice of $x,y\in V(D)$ is called antistrong. An antidirected trail which begins with a forward arc and ends with a backward arc is a forward-backward antidirected trail. For any $x,y\in V(D)$, if digraph D has a forward antidirected (x, y)-trail or a forward-backward antidirected (x, y)-trail, then D is weakly-antistrong. A digraph is anti-Eulerian if it admits a closed anti-directed Euler trail. Let $BC(Z_n, C)$ be a Bi-Circulant graph with $n\ge 3$ and $C=\{j_1, j_2, \ldots , j_{l}\}$, $l\ge 2$. This paper shows that the number of connected components of $BC(Z_n, C)$ is equal to $gcd(j_1-j_l, \ldots , j_{l-1}-j_{l}, n)$. Based on the result, necessary and sufficient conditions are given to decide whether a circulant digraph is antistrong, weakly-antistrong or anti-Eulerian.