Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Tuesday, October 20, 2015, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Nemanja Skoric

On the subgraphs of minimum degree 3

Given a graph $G$ on $n$ vertices and $2n − 1$ edges, what can we say about the smallest subset $S \in V(G)$ such that minimal degree of $G[S]$ is at least 3? It is easy to see that $G$ contains an induced subgraph with minimum degree at least 3, and Erdős, Faudree, Rousseau and Schelp showed that $|S| ≤ n − n^{1/2}$. Furthermore, Erdős conjectured that there exists an absolute constant $\epsilon > 0$ such that $|S| \leq (1 − \epsilon)n$. Note that $2n − 1$ is the smallest number of edges for which this question is sensible, since a wheel on $n$ vertices has $2n − 2$ edges and no proper induced subgraph with minimum degree 3. We make a small progress towards this conjecture. Since the problem apparently does not become easier when we restrict ourselves to bounded-degree graphs, here we show that if the maximum degree of $G$ is 5 then $|S| \leq n − n^{2/3}$.

If the description of the riddle did not get you interested, maybe the fact that Erdős mentioned this problem in his publication "Some of my favorite solved and unsolved problems in graph theory" will.

This is joint work with Ralph Keusch, Frank Mousset, Rajko Nenadov, Andreas Noever and Yury Person.

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