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DTSTART:20250309T070000
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DTSTART;TZID=America/Toronto:20251027T113000
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URL:https://uwaterloo.ca/combinatorics-and-optimization/events/algebraic-gr
 aph-theory-prangya-parida
SUMMARY:Algebraic Graph Theory-Prangya Parida
CLASS:PUBLIC
DESCRIPTION:TITLE: Cover-free Families on Graphs\n\nSPEAKER:\n Prangya Par
 ida\n\nAFFILIATION: \n\nUniversity of Ottawa\n\nLOCATION:\n Please contact
  Sabrina Lato for Zoom link.\n\nABSTRACT: A family of subsets of a t-se
 t is a d-cover-free family or\nd-CFF if no subset in the family is contain
 ed in the union of any d\nother subsets. Let t(d\, n) denote the minimum t
  for which there exists\na d-CFF of a t-set with n subsets. t(1\, n) is de
 termined using\nSperner’s theorem. For d ≥ 2\, we rely on bounds for t
 (d\, n).\nErdős\, Frankl\, and Füredi proved 3.106 log(n) < t(2\, n) < 
 5.512\nlog(n). \n\nA 2-CFF can be generalized by using a graph G with ver
 tices\ncorresponding to subsets in the set system. A G-CFF is a set system
 \nsuch that each edge of G specifies a pair of subsets not contained in\ne
 ach other and whose union must not contain any other subset. Let t(G)\nden
 ote the minimum t for which there exists a G-CFF. Thus\, t(K_n) =\nt(2\, n
 ).  \nIn this talk\, we discuss some classic results on cover-free famili
 es\,\nalong with general constructions of G-CFFs and specific construction
 s\nfor certain families of graphs. We show that for a graph G with n\nvert
 ices (no isolated vertices)\, t(1\, n) ≤ t(G) ≤ t(2\, n)\, and\nthat f
 or an infinite family of star graphs S_n with n vertices\, t(S_n)\n= t(1\,
  n). Interestingly\, we show how we can use a mixed-radix Gray\ncode to co
 nstruct CFFs on paths (P_n) and cycles (C_n) with n\nvertices. This leads 
 to the bound log(n) ≤ t(G) ≤ 1.89 log(n)\,\nwhere G is either P_n or C
 _n. \nThis is joint work with Lucia Moura.
DTSTAMP:20260402T152702Z
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