Let G be a graph and H be a hypergraph both on the same vertex set. We say that a hypergraph H is a Berge-G if there is a bijection f : E(G) → E(H) such that for e ∈ E(G) we have e ⊂ f (e). This generalizes the established definitions of "Berge path" and "Berge cycle" to general graphs. For a fixed graph G we examine the maximum possible size (i.e. the sum of the cardinality of each edge) of a hypergraph with no Berge-G as a subhypergraph. In the present paper we prove general bounds for this maximum when G is an arbitrary graph. We also consider the specific case when G is a complete bipartite graph and prove an analogue of the Kővári-Sós-Turán theorem. *
We present results on partitioning the vertices of $2$-edge-colored graphs
into monochromatic paths and cycles. We prove asymptotically the two-color case
of a conjecture of S\'ark\"ozy: the vertex set of every $2$-edge-colored graph
can be partitioned into at most $2\alpha(G)$ monochromatic cycles, where
$\alpha(G)$ denotes the independence number of $G$. Another direction, emerged
recently from a conjecture of Schelp, is to consider colorings of graphs with
given minimum degree. We prove that apart from $o(|V(G)|)$ vertices, the vertex
set of any $2$-edge-colored graph $G$ with minimum degree at least
$(1+\eps){3|V(G)|\over 4}$ can be covered by the vertices of two vertex
disjoint monochromatic cycles of distinct colors. Finally, under the assumption
that $\overline{G}$ does not contain a fixed bipartite graph $H$, we show that
in every $2$-edge-coloring of $G$, $|V(G)|-c(H)$ vertices can be covered by two
vertex disjoint paths of different colors, where $c(H)$ is a constant depending
only on $H$. In particular, we prove that $c(C_4)=1$, which is best possible
Fix graphs F and H and let ex(n, H, F ) denote the maximum possible number of copies of the graph H in an n-vertex F -free graph. The systematic study of this function was initiated by Alon and Shikhelman [J. Comb. Theory, B. 121 (2016)]. In this paper, we give new general bounds concerning this generalized Turán function. We also determine ex(n, P k , K 2,t ) (where P k is a path on k vertices) and ex(n, C k , K 2,t ) asymptotically for every k and t. For example, it is shown that for t ≥ 2 and k ≥ 5 we have ex(n, C k , K 2,t ) = 1 2k + o(1) (t − 1) k/2 n k/2 . We also characterize the graphs F that cause the function ex(n, C k , F ) to be linear in n. In the final section we discuss a connection between the function ex(n, H, F ) and so-called Berge hypergraphs.
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