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Michelle Delcourt

Publications and source records attributed to Michelle Delcourt.

28 records · Page 2Linked to original sources
arXiv2020

Progress towards Nash-Williams' Conjecture on Triangle Decompositions

Partitioning the edges of a graph into edge disjoint triangles forms a triangle decomposition of the graph. A famous conjecture by Nash-Williams from 1970 asserts that any sufficiently large, triangle divisible graph on $n$ vertices with minimum degree at least $0.75 n$ admits a triangle decomposition. In the light of recent results, the fractional version of this problem is of central importance. A fractional triangle decomposition is an assignment of non-negative weights to each triangle in a graph such that the sum of the weights along each edge is precisely 1. We show that for any graph on $n$ vertices with minimum degree at least $0.827327 n$ admits a fractional triangle decomposition. Combined with results of Barber, Kühn, Lo, and Osthus, this implies that for every sufficiently large triangle divisible graph on $n$ vertices with minimum degree at least $0.82733 n$ admits a triangle decomposition.

math.CO↗
arXiv2020

The Glauber dynamics for edge-colourings of trees

Let $T$ be a tree on $n$ vertices and with maximum degree $Δ$. We show that for $k\geq Δ+1$ the Glauber dynamics for $k$-edge-colourings of $T$ mixes in polynomial time in $n$. The bound on the number of colours is best possible as the chain is not even ergodic for $k \leq Δ$. Our proof uses a recursive decomposition of the tree into subtrees; we bound the relaxation time of the original tree in terms of the relaxation time of its subtrees using block dynamics and chain comparison techniques. Of independent interest, we also introduce a monotonicity result for Glauber dynamics that simplifies our proof.

cs.DS↗
arXiv2020

Independent Sets in Algebraic Hypergraphs

In this paper we study hypergraphs definable in an algebraically closed field. Our goal is to show, in the spirit of the so-called transference principles in extremal combinatorics, that if a given algebraic hypergraph is "dense" in a certain sense, then a generic low-dimensional subset of its vertices induces a subhypergraph that is also "dense." (For technical reasons, we only consider low-dimensional subsets that are parameterized by rational functions.) Our proof approach is inspired by the hypergraph containers method, developed by Balogh, Morris, and Samotij and independently by Saxton and Thomason (although adapting this method to the algebraic setting presents some unique challenges that do not occur when working with finite hypergraphs). Along the way, we establish a natural generalization of the classical dimension of fibers theorem in algebraic geometry, which is interesting in its own right.

math.CO↗
arXiv2018

Improved Bounds for Randomly Sampling Colorings via Linear Programming

A well-known conjecture in computer science and statistical physics is that Glauber dynamics on the set of $k$-colorings of a graph $G$ on $n$ vertices with maximum degree $Δ$ is rapidly mixing for $k\geΔ+2$. In FOCS 1999, Vigoda showed that the flip dynamics (and therefore also Glauber dynamics) is rapidly mixing for any $k>\frac{11}{6}Δ$. It turns out that there is a natural barrier at $\frac{11}{6}$, below which there is no one-step coupling that is contractive with respect to the Hamming metric, even for the flip dynamics. We use linear programming and duality arguments to fully characterize the obstructions to going beyond $\frac{11}{6}$. These extremal configurations turn out to be quite brittle, and in this paper we use this to give two proofs that the Glauber dynamics is rapidly mixing for any $k\ge\left(\frac{11}{6} - ε_0\right)Δ$ for some absolute constant $ε_0>0$. This is the first improvement to Vigoda's result that holds for general graphs. Our first approach analyzes a variable-length coupling in which these configurations break apart with high probability before the coupling terminates, and our other approach analyzes a one-step path coupling with a new metric that counts the extremal configurations. Additionally, our results extend to list coloring, a widely studied generalization of coloring, where the previously best known results required $k > 2 Δ$.

cs.DS↗
arXiv2018

A short nonalgorithmic proof of the containers theorem for hypergraphs

Recently the breakthrough method of hypergraph containers, developed independently by Balogh, Morris, and Samotij as well as Saxton and Thomason, has been used to study sparse random analogs of a variety of classical problems from combinatorics and number theory. The previously known proofs of the containers theorem use the so-called scythe algorithm---an iterative procedure that runs through the vertices of the hypergraph. (Saxton and Thomason have also proposed an alternative, randomized construction in the case of simple hypergraphs.) Here we present the first known deterministic proof of the containers theorem that is not algorithmic, i.e., it does not involve an iterative process. Our proof is less than 4 pages long while being entirely self-contained and conceptually transparent. Although our proof is completely elementary, it was inspired by considering hypergraphs in the setting of nonstandard analysis, where there is a notion of dimension capturing the logarithmic rate of growth of finite sets. Before presenting the proof in full detail, we include a one-page informal outline that refers to this notion of dimension and summarizes the essence of the argument.

math.CO↗
arXiv2018

Rapid mixing of Glauber dynamics for colorings below Vigoda's $11/6$ threshold

A well-known conjecture in computer science and statistical physics is that Glauber dynamics on the set of $k$-colorings of a graph $G$ on $n$ vertices with maximum degree $Δ$ is rapidly mixing for $k \geq Δ+2$. In FOCS 1999, Vigoda showed rapid mixing of flip dynamics with certain flip parameters on the set of proper $k$-colorings for $k > \frac{11}{6}Δ$, implying rapid mixing for Glauber dynamics. In this paper, we obtain the first improvement beyond the $\frac{11}{6}Δ$ barrier for general graphs by showing rapid mixing for $k > (\frac{11}{6} - η)Δ$ for some positive constant $η$. The key to our proof is combining path coupling with a new kind of metric that incorporates a count of the extremal configurations of the chain. Additionally, our results extend to list coloring, a widely studied generalization of coloring. Combined, these results answer two open questions from Frieze and Vigoda's 2007 survey paper on Glauber dynamics for colorings.

cs.DM↗
arXiv2018

Random 4-regular graphs have 3-star decompositions asymptotically almost surely

In 2006, Barat and Thomassen conjectured in 2006 that the edges of every planar 4-regular 4-edge-connected graph can be decomposed into copies of the star with 3 leaves. Shortly afterward, Lai constructed a counterexample to this conjecture. Using the small subgraph conditioning method of Robinson and Wormald, we prove that a random 4-regular graph has an $S_3$-decomposition asymptotically almost surely, provided the number of vertices is divisible by 3.

math.CO↗
arXiv2017

A combinatorial method for connecting BHV spaces representing different numbers of taxa

The phylogenetic tree space introduced by Billera, Holmes, and Vogtmann (BHV tree space) is a CAT(0) continuous space that represents trees with edge weights with an intrinsic geodesic distance measure. The geodesic distance measure unique to BHV tree space is well known to be computable in polynomial time, which makes it a potentially powerful tool for optimization problems in phylogenetics and phylogenomics. Specifically, there is significant interest in comparing and combining phylogenetic trees. For example, BHV tree space has been shown to be potentially useful in tree summary and consensus methods, which require combining trees with different number of leaves. Yet an open problem is to transition between BHV tree spaces of different maximal dimension, where each maximal dimension corresponds to the complete set of edge-weighted trees with a fixed number of leaves. We show a combinatorial method to transition between copies of BHV tree spaces in which trees with different numbers of taxa can be studied, derived from its topological structure and geometric properties. This method removes obstacles for embedding problems such as supertree and consensus methods in the BHV treespace framework.

q-bio.QM↗
arXiv2015

On a Conjecture of Thomassen

In 1989, Thomassen asked whether there is an integer-valued function f(k) such that every f(k)-connected graph admits a spanning, bipartite $k$-connected subgraph. In this paper we take a first, humble approach, showing the conjecture is true up to a log n factor.

math.CO↗
arXiv2015

Intersecting families of discrete structures are typically trivial

The study of intersecting structures is central to extremal combinatorics. A family of permutations $\mathcal{F} \subset S_n$ is \emph{$t$-intersecting} if any two permutations in $\mathcal{F}$ agree on some $t$ indices, and is \emph{trivial} if all permutations in $\mathcal{F}$ agree on the same $t$ indices. A $k$-uniform hypergraph is \emph{$t$-intersecting} if any two of its edges have $t$ vertices in common, and \emph{trivial} if all its edges share the same $t$ vertices. The fundamental problem is to determine how large an intersecting family can be. Ellis, Friedgut and Pilpel proved that for $n$ sufficiently large with respect to $t$, the largest $t$-intersecting families in $S_n$ are the trivial ones. The classic Erdős--Ko--Rado theorem shows that the largest $t$-intersecting $k$-uniform hypergraphs are also trivial when $n$ is large. We determine the \emph{typical} structure of $t$-intersecting families, extending these results to show that almost all intersecting families are trivial. We also obtain sparse analogues of these extremal results, showing that they hold in random settings. Our proofs use the Bollobás set-pairs inequality to bound the number of maximal intersecting families, which can then be combined with known stability theorems. We also obtain similar results for vector spaces.

math.CO↗