Author(s)

Hongjian Lai¹, Omaema Lasfar¹, Juan Liu²

¹Department of Mathematics, West Virginia University, Morgantown, USA.
²College of Big Data Statistics, Guizhou University of Finance and Economics, Guiyang, China.

A vertex cycle cover of a digraph H is a collection C = {C₁, C₂, …, C_k} of directed cycles in H such that these directed cycles together cover all vertices in H and such that the arc sets of these directed cycles induce a connected subdigraph of H. A subdigraph F of a digraph D is a circulation if for every vertex in F, the indegree of v equals its out degree, and a spanning circulation if F is a cycle factor. Define f (D) to be the smallest cardinality of a vertex cycle cover of the digraph obtained from D by contracting all arcs in F, among all circulations F of D. Adigraph D is supereulerian if D has a spanning connected circulation. In [International Journal of Engineering Science Invention, 8 (2019) 12-19], it is proved that if D₁ and D₂ are nontrivial strong digraphs such that D₁ is supereulerian and D₂ has a cycle vertex cover C’ with |C’| ≤ |V (D₁)|, then the Cartesian product D₁ and D₂ is also supereulerian. In this paper, we prove that for strong digraphs D₁ and D₂, if for some cycle factor F₁ of D₁, the digraph formed from D₁ by contracting arcs in F1 is hamiltonian with f (D₂) not bigger than |V (D₁)|, then the strong product D₁ and D₂ is supereulerian.

Supereulerian Digraph, Direct Product, Strong Product, Cycle Factors, Eulerian Digraph

Lai, H. , Lasfar, O. and Liu, J. (2021) Supereulerian Digraph Strong Products. Applied Mathematics, 12, 370-382. doi: 10.4236/am.2021.124026.