We construct infinitely many connected, circulant digraphs of outdegree three that have no hamiltonian circuit. All of our examples have an even number of vertices, and our examples are of two types: either every vertex in the digraph is adjacent to two diametrically opposite vertices, or every vertex is adjacent to the vertex diametrically opposite to itself.

Let D be a digraph. A subset S of V (D) is stable set if every pair vertices in non-adjacent D. collection disjoint paths path partition D, vertex exactly one . We say that and are orthogonal each contains S. digraph satisfies the α-property for maximum there exists such orthogonal. α-diperfect induced subdigraph α-property. In 1982, Berge proposed characterization digraphs terms forbidden anti...

The WL-rank of a digraph $$\Gamma$$ is defined to be the rank coherent configuration . WL-dimension smallest positive integer m for which identified by m-dimensional Weisfeiler–Leman algorithm. We classify Deza circulant graphs 4. In additional, it proved that each these has at most 3. Finally, we establish some families have 5 or 6 and

We show that the full automorphism group of a circulant digraph of square-free order is either the intersection of two 2-closed groups, each of which is the wreath product of 2-closed groups of smaller degree, or contains a transitive normal subgroup which is the direct product of two 2-closed groups of smaller degree. The work in this paper makes contributions to the solutions of two problems ...

Let D be the circulant digraph with n vertices and connection set {2, 3, c}. (Assume D is loopless and has outdegree 3.) Work of S.C. Locke and D.Witte implies that if n is a multiple of 6, c ∈ {(n/2) + 2, (n/2) + 3}, and c is even, then D does not have a hamiltonian cycle. For all other cases, we construct a hamiltonian cycle in D.

In this paper, we consider the asymptotic properties of the numbers of spanning trees and Eulerian trials in circulant digraphs and graphs. Let C(p, s1, s2, . . . , sk) be a directed or undirected circulant graph. Let T (C(p, s1, s2, . . . , sk)) and E(C(p, s1, s2, . . . , sk)) be the numbers ∗This work was partially supported by the Natural Science Foundation of China, email: fjzhang@jingxian....

An arc-colored digraph D is properly (properly-walk) connected if, for any ordered pair of vertices (u,v), the contains a directed path (a walk) from u to v such that arcs adjacent on (on have distinct colors. The proper connection number pc→(D) (the proper-walk wc→(D)) minimum colours make (properly-walk connected). We prove pc→(Cn(S))≤2 every circulant Cn(S) with S⊆{1,…,n−1},|S|≥2 and 1∈S. Fu...

The dichromatic number dc(D) of a digraph D is defined to be the minimum number of colors such that the vertices of D can be colored in such a way that every chromatic class induces an acyclic subdigraph in D. The cyclic circulant tournament is denoted by T = −→ C 2n+1(1, 2, . . . , n), where V (T ) = Z2n+1 and for every jump j ∈ {1, 2, . . . , n} there exist the arcs (a, a+ j) for every a ∈ Z2...

From a natural generalization to E* of the concept of congruence, it is possible to define a family of 2-regular digraphs that we call double commutative-step digraphs. They turn out to be Cayley diagrams of Abelian groups generated by two elements, and can be represented by L-shaped tiles which tessellate the plane periodically. A double commutative-step digraph with n vertices is said to be t...

In this paper we introduce a numerical invariant of digraphs which generalizes that of the number of connected components of a graph. The ao,clic disconnection ~(D) of a digraph D is the minimum number of (weakly) connected components of the subdigraphs obtained from D by deleting an acyclic set of arcs. We state some results about this invariant and compute its value for a variety of circulant...