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DTSTART:20220101T000000
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DTSTART;TZID=Asia/Seoul:20221215T100000
DTEND;TZID=Asia/Seoul:20221215T110000
DTSTAMP:20221209T153400
CREATED:20221109T130647Z
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UID:6454-1671098400-1671102000@dimag.ibs.re.kr
SUMMARY:Maya Sankar\, Homotopy and the Homomorphism Threshold of Odd Cycles
DESCRIPTION:Fix $r \ge 2$ and consider a family F of $C_{2r+1}$-free graphs\, each having minimum degree linear in its number of vertices. Such a family is known to have bounded chromatic number; equivalently\, each graph in F is homomorphic to a complete graph of bounded size. We disprove the analogous statement for homomorphic images that are themselves $C_{2r+1}$-free. Specifically\, we construct a family of dense $C_{2r+1}$-free graphs with no $C_{2r+1}$-free homomorphic image of bounded size. This provides the first nontrivial lower bound on the homomorphism threshold of longer odd cycles and answers a question of Ebsen and Schacht. \nOur proof relies on a graph-theoretic analogue of homotopy equivalence\, which allows us to analyze the relative placement of odd closed walks in a graph. This notion has surprising connections to the neighborhood complex\, and opens many further interesting questions.
URL:https://dimag.ibs.re.kr/event/2022-12-15/
LOCATION:Zoom ID: 224 221 2686 (ibsecopro)
CATEGORIES:Virtual Discrete Math Colloquium
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