Description
Chair: Heesung Shin
The study of geometric transversals deals with generalizations of Helly's theorem. Instead of intersecting convex sets by points, we now wish to intersect convex sets by $k$-dimensional affine flats. The modern development of Helly type theorems (colorful, fractional, $(p,q)$ versions) has given rise to a number of new questions about general geometric transversals. In this talk I will survey...
A graph class $\mathcal{G}$ has the strong EH(Erd\H{o}s-Hajnal) property if there is a constant $c=c(\mathcal{G}) > 0$ such that for every member $G$ of $\mathcal{G}$, either $G$ or its complement has $K_{m, m}$ as a subgraph where $m \geq c \left| V(G) \right|$. We prove that the chordal graphs satisfy strong Erd\H{o}s-Hajnal property with constant $c = 2/9$.
On the other hand, a...
In 1952, Dirac proved that every $2$-connected $n$-vertex graph with the minimum degree $k+1$ contains a cycle of length at least $\min\{n, 2(k+1)\}$. Here we obtain a stability version of this result by characterizing those graphs with minimum degree $k$ and circumference at most $2k+1$.
We present applications of the above-stated result by obtaining generalized Tur\'an numbers.
In...
