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SUMMARY:Counting cycles in planar graphs - Ryan Martin (Iowa State)
DTSTART:20260205T143000Z
DTEND:20260205T153000Z
UID:TALK243865@talks.cam.ac.uk
CONTACT:103978
DESCRIPTION:Basic Tur\\'an theory asks how many edges a graph can have\, g
 iven certain restrictions such as not having a large clique. A more genera
 lized Tur\\'an question asks how many copies of a fixed subgraph $H$ the g
 raph can have\, given certain restrictions. There has been a great deal of
  recent interest in the case where the restriction is planarity. In this t
 alk\, we will discuss some of the general results in the field\, primarily
  the asymptotic value of ${\\bf N}_{\\mathcal P}(n\,H)$\, which denotes th
 e maximum number of copies of $H$ in an $n$-vertex planar graph. In partic
 ular\, we will focus on the case where $H$ is a cycle.\n\n \n\nIt was dete
 rmined that ${\\bf N}_{\\mathcal P}(n\,C_{2m})=(n/m)^m+o(n^m)$ for small v
 alues of $m$ by Cox and Martin and resolved for all $m$ by Lv\, Gy\\H{o}ri
 \, He\, Salia\, Tompkins\, and Zhu.\n\nThe case of $H=C_{2m+1}$ is more di
 fficult and it is conjectured that ${\\bf N}_{\\mathcal P}(n\,C_{2m+1})=2m
 (n/m)^m+o(n^m)$. \n\n \n\nWe will discuss recent progress on this problem\
 , including verification of the conjecture in the case where $m=3$ and $m=
 4$ and a lemma which reduces the solution of this problem for any $m$ to a
  so-called ``maximum likelihood'' problem. The maximum likelihood problem 
 is\, in and of itself\, an interesting question in random graph theory.
LOCATION:MR12
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