How connectivity affects the extremal number of trees

Abstract

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 $(1/4+o_k(1))nk$.

Figures (2)

• Figure 1: The graph $\textsf{G}_{n,k,s}$ on the left, the graph $\textsf{S}_{n,x}$ in the middle and the graph $\textsf{P}_{n,x}$ on the right.
• Figure 2: The graph $T_k$ in $\textsf{S}_{n,x_1}$ on the left and the graph $T_k$ on the right.

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 2.1
• proof
• Theorem 2.2: Woodall woodall1976maximal, Kopylov kopylov1977maximal
• Theorem 2.3: Kopylov kopylov1977maximal, Woodall woodall1976maximal, Fan, Lv and Wang fan2004cycles
• Claim 3.1
• proof : Proof of claim
• Claim 3.2