On the size of planarly connected crossing graphs

We prove that if an $n$-vertex graph $G$ can be drawn in the plane such that each pair of crossing edges is independent and there is a crossing-free edge that connects their endpoints, then $G$ has $O(n)$ edges. Graphs that admit such drawings are related to quasi-planar graphs and to maximal $1$-planar and fan-planar graphs.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

The biased odd cycle game

In this paper we consider biased Maker-Breaker games played on the edge set of a given graph $G$. We prove that for every $\delta>0$ and large enough $n$, there exists a constant $k$ for which if $\delta(G)\geq \delta n$ and $\chi(G)\geq k$, then Maker can build an odd cycle in the $(1:b)$ game for $b=O(\frac{n}{\log^2 n})$. We also consider the analogous game where Maker and Breaker claim vertices instead of edges. This is a special case of the following well known and notoriously difficult problem due to Duffus, {\L}uczak and R\"{o}dl: is it true that for any positive constants $t$ and $b$, there exists an integer $k$ such that for every graph $G$, if $\chi(G)\geq k$, then Maker can build a graph which is not $t$-colorable, in the $(1:b)$ Maker-Breaker game played on the vertices of $G$?Comment: 10 page

Asyrnptotics for the Turan number of Berge-K-2,K-t

Let F be a graph. A hypergraph is called Berge-F if it can be obtained by replacing each edge in F by a hyperedge containing it.Let F be a family of graphs. The Turan number of the family Berge-F is the maximum possible number of edges in an r-uniform hypergraph on n vertices containing no Berge-F as a subhypergraph (for every F is an element of F) and is denoted by ex(r)(n, F).We determine the asymptotics for the Turan number of Berge-K-2,K-t, by showingex(3)(n, K-2,K-t) = (1 + o(1))1/6 (t - 1)(3/2) . n(3/2)for any given t >= 7. We study the analogous question for linear hypergraphs and showex(3)(n,{C-2, K-2,K-t}) = (1 + o(t)(1))1/6 root t - 1. n(3/2).We also prove general upper and lower bounds on the Turan numbers of a class of graphs including ex(r)(n, K-2,K-t), ex(r)(n, {C-2, K-2,K-t}), and ex,(n, C-2k) for r >= 3. Our bounds improve the results of Gerbner and Palmer [18], Fiiredi and Ozkahya [15], Timmons [37], and provide a new proof of a result of Jiang and Ma [26]. (C) 2019 Elsevier Inc. All rights reserved

On clique coverings of complete multipartite graphs

A clique covering of a graph G is a set of cliques of G such that any edge of G is contained in one of these cliques, and the weight of a clique covering is the sum of the sizes of the cliques in it. The sigma clique cover number scc(G) of a graph G, is defined as the smallest possible weight of a clique covering of G.Let K-t(d) denote the complete t-partite graph with each part of size d. We prove that for any fixed d >= 2, we havelim(t ->infinity) scc(K-t(d)) = d/2t log t.This disproves a conjecture of Davoodi et al. (2016). (C) 2019 Elsevier B.V. All rights reserved

On The Ratio Of Maximum And Minimum Degree In Maximal Intersecting Families

To study how balanced or unbalanced a maximal intersecting family F subset of ((vertical bar n vertical bar) (r)) is we consider the ratio R(F) = Delta(F)/delta(F) of its maximum and minimum degree. We determine the order of magnitude of the function m(n, r), the minimum possible value of R(F), and establish some lower and upper bounds on the function M(n, r), the maximum possible value of R(F). To obtain constructions that show the bounds on m(n, r) we use a theorem of Blokhuis on the minimum size of a non-trivial blocking set in projective planes. (C) 2012 Elsevier B.V. All rights reserved.WoSScopu

Stability results for vertex Turan problems in Kneser graphs

The vertex set of the Kneser graph K(n, k) is V = (([n])(k)) and two vertices are adjacent if the corresponding sets are disjoint. For any graph F, the largest size of a vertex set U subset of V such that K(n, k)[U] is F-free, was recently determined by Alishahi and Taherkhani, whenever n is large enough compared to k and F. In this paper, we determine the second largest size of a vertex set W subset of V such that K(n, k)[W] is F-free, in the case when F is an even cycle or a complete multi-partite graph. In the latter case, we actually give a more general theorem depending on the chromatic number of F