Monochromatic connected matchings in 2-edge-colored multipartite graphs

A matching $M$ in a graph $G$ is connected if all the edges of $M$ are in the same component of $G$. Following \L uczak,there have been many results using the existence of large connected matchings in cluster graphs with respect to regular partitions of large graphs to show the existence of long paths and other structures in these graphs. We prove exact Ramsey-type bounds on the sizes of monochromatic connected matchings in $2$-edge-colored multipartite graphs. In addition, we prove a stability theorem for such matchings.Comment: 29 pages, 2 figure

Conditions for a bigraph to be super-cyclic

A hypergraph $\mathcal H$ is super-pancyclic if for each $A \subseteq V(\mathcal H)$ with $|A| \geq 3$, $\mathcal H$ 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 for this. In particular, they are sufficient for every hypergraph $\mathcal H$ with $\delta(\mathcal H)\geq \max\{|V(\mathcal H)|, \frac{|E(\mathcal H)|+10}{4}\}$. We also consider super-cyclic bipartite graphs: those are $(X,Y)$-bigraphs $G$ such that for each $A \subseteq X$ with $|A| \geq 3$, $G$ 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.Comment: 17 page