for Cycles of Even Length

Carnegie Mellon University
18th June 2015

Joint work with Boris Bukh.

Incomplete history

$\operatorname{ex}(n, F)$ – largest # of edges in an $F$-free graph

Erdős–Stone–Simonovits Theorem

$$\operatorname{ex}(n, F) = \left(1 - \frac{1}{\chi - 1}\right){n \choose 2} + o(n^2)$$

Degenerate case $\chi = 2$

• Complete bipartite $K_{s,t}$
• Cycle of even length $C_{2k}$

Even cycles

Erdős: $\operatorname{ex}(n, C_{2k}) \le$ $\gamma_k$ $n^{1+1/k}$

 1974 Bondy–Simonovits $20k$ 2000 Verstraëte $8(k-1)$ 2012 Pikhurko $k-1$ 2014 Bukh–J. $80\sqrt{k\log k}$

Plan of talk

 ▶ Revisit Pikhurko's proof How to make more math?

 If $V(\Theta) =$ $A$ $\cup$ $B$ is not bipartition Then Path from $A$ to $B$ of any possible length

Pikhurko's proof

Suppose $G$ is $C_{2k}$-free and $e(G) \ge n^{1+1/k}$

Take $H\subset G$ of minimum degree $\delta(H) \ge e(G) / n \ge n^{1/k}$

 Claim $H_i$ is $\Theta$-free for $i < k$

Proof of the claim

Find $y$ the rightmost common ancestor of $\Theta \cap V_i$ in $T$.

Pick $z$ a child of $y$, and find $C_{2k}$ using $\Theta$.

Pikhurko's proof (cont.)

 Obs Bipartite graph of min deg $\ge k$ contains $\Theta$-graph. Cor Average degree of $H_i \le 2k$.

Punchline
$\lesssim 2k$
$V_{i}$
$\lesssim 2k$
$\gtrsim \delta$
$V_{i+1}$

$\frac{v_{i+1}}{v_i}\lesssim \frac{\delta}{k}$$\implies \left(\frac{\delta}{k}\right)^k \lesssim n$$\implies e(G) \lesssim k\ n^{1+1/k}$

Plan of talk

 ▶ Review Pikhurko's proof ▶ How to make more math?

Trilayered graph

1. Build breath-first search tree
2. Find degree conditions for $\Theta$-free trilayered graph
3. Find $C_{2k}$ if trilayered graph contains $\Theta$-graph

Minimum degree condition

Dichotomy
 Large minimum degree Implies Denser trilayered subgraph Or $\Theta$-graph on two levels

Maximum degree condition

 Question Is $Bg$ big? Yes $V_{i+1} = Bg \cup Sm$ No $V_{i+1} = Sm$

Carnegie Mellon University
zj@cmu.edu