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SUMMARY:10 avril 2025: Daria Pchelina et François Pirot
DTSTART:20250410T120000Z
DTEND:20250410T150000Z
DTSTAMP:20260611T011200Z
UID:indico-event-14194@indico.math.cnrs.fr
DESCRIPTION:Speakers: Daria Pchelina (CNRS\, ENS Lyon)\, Francois Pirot (U
 niversité Paris Saclay)\n\n14h00  Daria Pchelina  CNRS\, ENS de Lyon\nOpt
 imal disc and sphere packings\nHow can we arrange an infinite number of sp
 heres to maximize the proportion of space they occupy? Kepler conjectured 
 that the "cannonball" packing is the optimal way to do it. This conjecture
  remained unproved for almost 400 years until Hales and Ferguson provided 
 a 250-page proof accompanied by hundreds of thousands of lines of computer
  code. Given an infinite number of coins of three fixed radii\, how can we
  place them on the plane to maximize the proportion of the covered surface
 ? A disc packing is called triangulated if its contact graph is triangulat
 ed. We identified optimal packings for several triplets of disc sizes\, al
 l of which are triangulated. Conversely\, we also showed that for certain 
 other triplets\, no triangulated packing is optimal. Building on our exper
 tise in multi-size disc packings\, we extend our research to two-sphere pa
 ckings. Simplicial sphere packings are those whose contact graphs form pur
 e simplicial 3-complexes. We consider the only ratio of sphere sizes that 
 allows such packings which are conjectured to be optimal.\n \n\n\n15h30  
 François Pirot  Université Paris-Saclay\nFractional Domatic Number and M
 inimum Degree\nGiven a graph G\, a dominating set of G is a set X⊆V(G) s
 uch that N[X]=V(G). Dominating sets are often used to model monitoring pro
 blems in networks. A classical problem in graph theory is to find the mini
 mum size γ(G) of a dominating set of G\, called the domination number of 
 G. A possible approach to this problem is to study a stronger parameter\, 
 the domatic number of G\, which is the maximum number of pairwise disjoint
  dominating sets of G. In this talk\, I will present the fractional relaxa
 tion of this parameter\, the fractional domatic number\, and study its ext
 remal value with respect to the minimum degree of G. I will present an asy
 mptotically tight bound\, and then focus on graphs G of minimum degree 2\,
  showing that they all have fractional domatic number at least 5/2 unless 
 at least one of 8 bad graphs appears as an isolated component in G. This i
 s joint work with Quentin Chuet\, Hugo Demaret\, and Hoang La.\n\n\n\nhttp
 s://indico.math.cnrs.fr/event/14194/
URL:https://indico.math.cnrs.fr/event/14194/
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