기초과학연구원 이산수학그룹
On May 30, 2023, Suyun Jiang (江素云) from Jianghan University gave a talk on the maximum number of edges in a connected graph without a fixed tree at the Discrete Math Seminar. She is currently a visiting scholar at the IBS Extremal Combinatorics and Probability Group. The title of her talk was “How connectivity affects the extremal number of trees.”
The Erdős-Sós conjecture states that the maximum number of edges in an $n$-vertex graph without a given $k$-vertex tree is at most $\frac {n(k-2)}{2}$. Despite significant interest, the conjecture remains unsolved. Recently, Caro, Patkós, and Tuza considered this problem for host graphs that are connected. Settling a problem posed by them, for a $k$-vertex tree $T$, we construct $n$-vertex connected graphs that are $T$-free with at least $(1/4-o_k(1))nk$ edges, showing that the additional connectivity condition can reduce the maximum size by at most a factor of 2. Furthermore, we show that this is optimal: there is a family of $k$-vertex brooms $T$ such that the maximum size of an $n$-vertex connected $T$-free graph is at most $(1/4+o_k(1))nk$.
The IBS discrete mathematics group welcomes Dr. Suyun Jiang (江素云), a new visiting scholar from Jianghan University, China. She is visiting the IBS extremal combinatorics and probability group for 21 months from November 4, 2022. She received her Ph.D. from Shandong University in China and is currently an assistant researcher in the School of Artificial Intelligence, Jianghan University, Wuhan, China.