Three Types of Two-Disjoint-Cycle-Cover Pancyclicity and Their Applications to Cycle Embedding in Locally Twisted Cubes

A graph $G=(V,E)$ is two-disjoint-cycle-cover $[r_1,r_2]$-pancyclic if for any integer $l$ satisfying $r_1 \leq l \leq r_2$, there exist two vertex-disjoint cycles $C_1$ and $C_2$ in $G$ such that the lengths of $C_1$ and $C_2$ are $l$ and $|V(G)| - l$, respectively, where $|V(G)|$ denotes the total number of vertices in $G$. On the basis of this definition, we further propose Ore-type conditions for graphs to be two-disjoint-cycle-cover vertex/edge $[r_1,r_2]$-pancyclic. In addition, we study cycle embedding in the $n$-dimensional locally twisted cube $LTQ_n$ under the consideration of two-disjoint-cycle-cover vertex/edge pancyclicity.

Enumeration of Unlabeled Directed Hypergraphs

We consider the enumeration of unlabeled directed hypergraphs by using Pólya's counting theory and Burnside's counting lemma. Instead of characterizing the cycle index of the permutation group acting on the hyperarc set $A$, we treat each cycle in the disjoint cycle decomposition of a permutation $\rho$ acting on $A$ as an equivalence class (or orbit) of $A$ under the operation of the group generated by $\rho$. Compared to the cycle index method, our approach is more effective in dealing with the enumeration of directed hypergraphs. We deduce the explicit counting formulae for the unlabeled $q$-uniform and unlabeled general directed hypergraphs. The former generalizes the well known result for 2-uniform directed hypergraphs, i.e., for the ordinary directed graphs introduced by Harary and Palmer.