Conditions for a Bigraph to be Super-Cyclic

Abstract

A hypergraph $\mathcal H$ is super-pancyclic if for each $A \subseteq V(\mathcal H)$ with $|A| \geqslant 3$, $\mathcal H$ contains a Berge cycle with base vertex set $A$. We present two natural necessary conditions for a hypergraph to be super-pancyclic, and show that in several classes of hypergraphs these necessary conditions are also sufficient. In particular, they are sufficient for every hypergraph $\mathcal H$ with $ \delta(\mathcal H)\geqslant \max\{|V(\mathcal H)|, \frac{|E(\mathcal H)|+10}{4}\}$.
 We also consider super-cyclic bipartite graphs: those are $(X,Y)$-bigraphs $G$ such that for each $A \subseteq X$ with $|A| \geqslant 3$, $G$ has a cycle $C_A$ such that $V(C_A)\cap X=A$. Such graphs are incidence graphs of super-pancyclic hypergraphs, and our proofs use the language of such graphs.

Pósa-Type Results for Berge Hypergraphs

“…For a hypergraph H consider the incidence bipartite graph G(A, B), where the vertices in A represent vertices of H and the vertices in B represent hyperedges of H. A vertex a ∈ A is adjacent with a vertex b ∈ B in G if and only if the vertex in H corresponding to a is contained in the hyperedge corresponding to b in H. There is a one-to-one correspondence between cycles in G and Berge cycles of H. The corresponding cycle in G of a Hamiltonian Berge cycle in H is a cycle containing all vertices of the set A. In [17,18,25,21,22] is related work on cycles covering color classes in bipartite graphs. In the coming subsections, we present the motivation for this work and introduce some necessary definitions and notions.…”

Pósa-type results for Berge-hypergraphs

“…Note that, incidence bipartite graphs of simple hypergraphs have an extra condition-the neighborhoods of any two vertices representing different hyperedges of the hypergraph are distinct. While the topic of searching for cycles covering a color class for the class of all bipartite graphs is an interesting and active topic of research see [15,16,21,19,18], in this paper, we study simple hypergraphs only.…”