On October 10, 2019, Prof. Alexandr V. Kostochka from the University of Illinois at Urbana-Champaign gave a talk at the discrete math seminar. The title of his talk was “Reconstructing graphs from smaller subgraphs“.
On October 8, 2019, Prof. Alexandr V. Kostochka from the University of Illinois at Urbana-Champaign gave a colloquium talk at the KAIST Mathematical Sciences – IBS Discrete Mathematics Group Joint Colloquium. The title of his talk was “On Ramsey-type problems for paths and cycles in dense graphs“. He will stay one week at IBS Discrete Mathematics Group.
A graph or graph property is $\ell$-reconstructible if it is determined by the multiset of all subgraphs obtained by deleting $\ell$ vertices. Apart from the famous Graph Reconstruction Conjecture, Kelly conjectured in 1957 that for each $\ell\in\mathbb N$, there is an integer $n=n(\ell)$ such that every graph with at least $n$ vertices is $\ell$-reconstructible.
We show that for each $n\ge7$ and every $n$-vertex graph $G$, the degree list and connectedness of $G$ are $3$-reconstructible, and the threshold $n\geq 7$ is sharp for both properties. We also show that all $3$-regular graphs are $2$-reconstructible.
A well-known Ramsey-type puzzle for children is to prove that among any 6 people either there are 3 who know each other or there are 3 who do not know each other. More generally, a graph $G$ arrows a graph $H$ if for any coloring of the edges of $G$ with two colors, there is a monochromatic copy of $H$. In these terms, the above puzzle claims that the complete $6$-vertex graph $K_6$ arrows the complete $3$-vertex graph $K_3$.
We consider sufficient conditions on the dense host graphs $G$ to arrow long paths and even cycles. In particular, for large $n$ we describe all multipartite graphs that arrow paths and cycles with $2n$ edges. This implies a conjecture by Gyárfás, Ruszinkó, Sárkőzy and Szemerédi from 2007 for such $n$. Also for large $n$ we find which minimum degree in a $(3n-1)$-vertex graph $G$ guarantees that $G$ arrows the $2n$-vertex path. This yields a more recent conjecture of Schelp.
This is joint work with Jozsef Balogh, Mikhail Lavrov and Xujun Liu.