A Bound on Turán Number

for Cycles of Even Length


姜子麟 Zilin Jiang
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?
images from boltcity.com

Combinatorial gadget

$\Theta$-graph (at least $2k$ vertices)
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?
images from boltcity.com

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

Well-placed $\Theta$-graph

Minimum degree condition

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

Maximum degree condition

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

姜子麟 Zilin Jiang
Carnegie Mellon University
zj@cmu.edu

images from boltcity.com