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Jean-Sebastien Sereni

Publications and source records attributed to Jean-Sebastien Sereni.

5 recordsLinked to original sources
arXiv2013

Extensions of Fractional Precolorings show Discontinuous Behavior

We study the following problem: given a real number k and integer d, what is the smallest epsilon such that any fractional (k+epsilon)-precoloring of vertices at pairwise distances at least d of a fractionally k-colorable graph can be extended to a fractional (k+epsilon)-coloring of the whole graph? The exact values of epsilon were known for k=2 and k\ge3 and any d. We determine the exact values of epsilon for k \in (2,3) if d=4, and k \in [2.5,3) if d=6, and give upper bounds for k \in (2,3) if d=5,7, and k \in (2,2.5) if d=6. Surprisingly, epsilon viewed as a function of k is discontinuous for all those values of d.

math.CO↗
arXiv2012

A new lower bound based on Gromov's method of selecting heavily covered points

Boros and Furedi (for d=2) and Barany (for abritrary d) proved that there exists a positive real number c_d such that for every set P of n points in R^d in general position, there exists a point of R^d contained in at least c_d n!/(d+1)!(n-d-1)! d-simplices with vertices at the points of P. Gromov improved the lower bound on c_d by topological means. Using methods from extremal combinatorics, we improve one of the quantities appearing in Gromov's approach and thereby provide a new stronger lower bound on c_d for arbitrary d. In particular, we improve the lower bound on c_3 from 0.06332 to more than 0.07480; the best upper bound known on c_3 being 0.09375.

math.CO↗
arXiv2011

Min-max relations for odd cycles in planar graphs

Let m(G) be the maximum number of vertex-disjoint odd cycles of a graph G and t(G) the minimum number of vertices whose removal makes G bipartite. We show that t(G)<=6m(G) if G is planar. This improves the previous bound t(G)<=10m(G) by Fiorini, Hardy, Reed and Vetta [Math. Program. Ser. B 110 (2007), 71-91].

math.CO↗
arXiv2009

The last fraction of a fractional conjecture

Reed conjectured that for every $\varepsilon>0$ and every integer $Δ$, there exists $g$ such that the fractional total chromatic number of every graph with maximum degree $Δ$ and girth at least $g$ is at most $Δ+1+\varepsilon$. The conjecture was proven to be true when $Δ=3$ or $Δ$ is even. We settle the conjecture by proving it for the remaining cases.

math.CO↗