arXiv subjects

Jean-Sébastien Sereni

Publications and source records attributed to Jean-Sébastien Sereni.

At least 19 recordsLinked to original sources

arXiv2026

A relaxation of the Bermond-Thomassen conjecture

The well-known Bermond-Thomassen conjecture states that every digraph of minimum out-degree at least $2k-1$ contains $k$ vertex-disjoint directed cycles. Despite being posed in 1981, this conjecture remains unresolved for all $k \ge 4$. We prove a relaxation of this conjecture: every digraph $D$ of minimum out-degree at least $2k-1$ contains $k$ vertex-disjoint cycles, each of which either is directed or can be made directed by reversing one of its arcs. This bound is sharp and answers a question raised by Cames van Batenburg during the online workshop "Entropy Compression and Related Methods" in $2021$.

math.CO↗

arXiv2025

On the irreducibility and monodromy of Tutte polynomials

We study algebraic properties of the Tutte polynomial of a matroid and its generalizations to other combinatorially defined bivariate polynomial invariants. Merino, de Mier and Noy showed that the Tutte polynomial of a connected matroid is irreducible, and Bohn, Cameron and M{ü}ller conjectured the stronger property that the Galois/monodromy group of the Tutte polynomial of a connected matroid of rank r is isomorphic to the full symmetric group on r letters. First, we generalize the result of Merino-de Mier-Noy to the context of general ranked sets by exploiting a recent translation of the Brylawski relations, satisfied by the coefficients of the Tutte polynomial, into a functional identity. Second, we give the first confirmation of the conjecture of Bohn-Cameron-M{ü}ller for infinite families of connected matroids, including the cycle graphs and the uniform matroids. Moreover, we apply the large sieve to obtain a probabilistic statement showing that suitable linear combinations of coprime Tutte polynomials generically satisfy the conjecture.

math.CO↗

arXiv2021

Fractional chromatic number, maximum degree and girth

We introduce a new method for computing bounds on the independence number and fractional chromatic number of classes of graphs with local constraints, and apply this method in various scenarios. We establish a formula that generates a general upper bound for the fractional chromatic number of triangle-free graphs of maximum degree~$Δ\ge 3$. This upper bound matches that deduced from the fractional version of Reed's bound for small values of~$Δ$, and improves it when~$Δ\ge 17$, transitioning smoothly to the best possible asymptotic regime, barring a breakthrough in Ramsey theory. Focusing on smaller values of~$Δ$, we also demonstrate that every graph of girth at least~$7$ and maximum degree~$Δ$ has fractional chromatic number at most~$1+ \min_{k \in \mathbb{N}} \frac{2Δ+ 2^{k-3}}{k}$. In particular, the fractional chromatic number of a graph of girth~$7$ and maximum degree~$Δ$ is at most~$\frac{2Δ+9}{5}$ when~$Δ\in [3,8]$, at most~$\frac{Δ+7}{3}$ when~$Δ\in [8,20]$, at most~$\frac{2Δ+23}{7}$ when~$Δ\in [20,48]$, and at most~$\fracΔ{4}+5$ when~$Δ\in [48,112]$. In addition, we also obtain new lower bounds on the independence ratio of graphs of maximum degree~$Δ\in \{3,4,5\}$ and girth~$g\in \{6,\dotsc,12\}$, notably~$1/3$ when~$(Δ,g)=(4,10)$ and~$2/7$ when~$(Δ,g)=(5,8)$.

math.CO↗

arXiv2020

An algorithmic framework for colouring locally sparse graphs

We develop an algorithmic framework for graph colouring that reduces the problem to verifying a local probabilistic property of the independent sets. With this we give, for any fixed $k\ge 3$ and $\varepsilon>0$, a randomised polynomial-time algorithm for colouring graphs of maximum degree $Δ$ in which each vertex is contained in at most $t$ copies of a cycle of length $k$, where $1/2\le t\le Δ^\frac{2\varepsilon}{1+2\varepsilon}/(\logΔ)^2$, with $\lfloor(1+\varepsilon)Δ/\log(Δ/\sqrt t)\rfloor$ colours. This generalises and improves upon several notable results including those of Kim (1995) and Alon, Krivelevich and Sudakov (1999), and more recent ones of Molloy (2019) and Achlioptas, Iliopoulos and Sinclair (2019). This bound on the chromatic number is tight up to an asymptotic factor $2$ and it coincides with a famous algorithmic barrier to colouring random graphs.

cs.DS↗

arXiv2020

Graph structure via local occupancy

The first author together with Jenssen, Perkins and Roberts (2017) recently showed how local properties of the hard-core model on triangle-free graphs guarantee the existence of large independent sets, of size matching the best-known asymptotics due to Shearer (1983). The present work strengthens this in two ways: first, by guaranteeing stronger graph structure in terms of colourings through applications of the Lovász local lemma; and second, by extending beyond triangle-free graphs in terms of local sparsity, treating for example graphs of bounded local edge density, of bounded local Hall ratio, and of bounded clique number. This generalises and improves upon much other earlier work, including that of Shearer (1995), Alon (1996) and Alon, Krivelevich and Sudakov (1999), and more recent results of Molloy (2019), Bernshteyn (2019) and Achlioptas, Iliopoulos and Sinclair (2019). Our results derive from a common framework built around the hard-core model. It pivots on a property we call local occupancy, giving a clean separation between the methods for deriving graph structure with probabilistic information and verifying the requisite probabilistic information itself.

math.CO↗

arXiv2020

Revisiting a theorem by Folkman on graph colouring

We give a short proof of the following theorem due to Jon H. Folkman (1969): The chromatic number of any graph is at most $2$ plus the maximum over all subgraphs of the difference between half the number of vertices and the independence number.

math.CO↗

arXiv2020

Bipartite independence number in graphs with bounded maximum degree

We consider a natural, yet seemingly not much studied, extremal problem in bipartite graphs. A bi-hole of size $t$ in a bipartite graph $G$ is a copy of $K_{t, t}$ in the bipartite complement of $G$. Let $f(n, Δ)$ be the largest $k$ for which every $n \times n$ bipartite graph with maximum degree $Δ$ in one of the parts has a bi-hole of size $k$. Determining $f(n, Δ)$ is thus the bipartite analogue of finding the largest independent set in graphs with a given number of vertices and bounded maximum degree. Our main result determines the asymptotic behavior of $f(n, Δ)$. More precisely, we show that for large but fixed $Δ$ and $n$ sufficiently large, $f(n, Δ) = Θ(\frac{\log Δ}Δ n)$. We further address more specific regimes of $Δ$, especially when $Δ$ is a small fixed constant. In particular, we determine $f(n, 2)$ exactly and obtain bounds for $f(n, 3)$, though determining the precise value of $f(n, 3)$ is still open.

math.CO↗

arXiv2020

On the effect of symmetry requirement for rendezvous on the complete graph

We consider a classic rendezvous game where two players try to meet each other on a set of $n$ locations. In each round, every player visits one of the locations and the game finishes when the players meet at the same location. The goal is to devise strategies for both players that minimize the expected waiting time till the rendezvous. In the asymmetric case, when the strategies of the players may differ, it is known that the optimum expected waiting time of $\frac{n+1}{2}$ is achieved by the wait-for-mommy pair of strategies, where one of the players stays at one location for $n$ rounds, while the other player searches through all the $n$ locations in a random order. However, if we insist that the players are symmetric --- they are expected to follow the same strategy --- then the best known strategy, proposed by Anderson and Weber, achieves an asymptotic expected waiting time of $0.829 n$. We show that the symmetry requirement indeed implies that the expected waiting time needs to be asymptotically larger than in the asymmetric case. Precisely, we prove that for every $n\geqslant 2$, if the players need to employ the same strategy, then the expected waiting time is at least $\frac{n+1}{2}+\varepsilon n$, where $\varepsilon=2^{-36}$.

cs.GT↗

arXiv2019

A $4$-choosable graph that is not $(8:2)$-choosable

In 1980, Erdős, Rubin and Taylor asked whether for all positive integers $a$, $b$, and $m$, every $(a:b)$-choosable graph is also $(am:bm)$-choosable. We provide a negative answer by exhibiting a $4$-choosable graph that is not $(8:2)$-choosable.

math.CO↗

arXiv2019

On fractional fragility rates of graph classes

We consider, for every positive integer $a$, probability distributions on subsets of vertices of a graph with the property that every vertex belongs to the random set sampled from this distribution with probability at most $1/a$. Among other results, we prove that for every positive integer~$a$ and every planar graph $G$, there exists such a probability distribution with the additional property that deleting the random set creates a graph with component-size at most $(Δ(G)-1)^{a+O(\sqrt{a})}$, or a graph with treedepth at most $O(a^3\log_2(a))$. We also provide nearly-matching lower bounds.

math.CO↗

arXiv2019

Variations on the Petersen colouring conjecture

The Petersen colouring conjecture states that every bridgeless cubic graph admits an edge-colouring with $5$ colours such that for every edge $e$, the set of colours assigned to the edges adjacent to $e$ has cardinality either $2$ or $4$, but not $3$. We prove that every bridgeless cubic graph $G$ admits an edge-colouring with $4$ colours such that at most $\frac45\cdot|V(G)|$ edges do not satisfy the above condition. This bound is tight and the Petersen graph is the only connected graph for which the bound cannot be decreased. We obtain such a $4$-edge-colouring by using a carefully chosen subset of edges of a perfect matching, and the analysis relies on a simple discharging procedure with essentially no reductions and very few rules.

math.CO↗

arXiv2019

On generalized choice and coloring numbers

A well-known result of Alon shows that the coloring number of a graph is bounded by a function of its choosability. We explore this relationship in a more general setting with relaxed assumptions on color classes, encoded by a graph parameter.

math.CO↗

arXiv2018

Limits of Order Types

We apply ideas from the theory of limits of dense combinatorial structures to study order types, which are combinatorial encodings of finite point sets. Using flag algebras we obtain new numerical results on the Erdős problem of finding the minimal density of 5-or 6-tuples in convex position in an arbitrary point set, and also an inequality expressing the difficulty of sampling order types uniformly. Next we establish results on the analytic representation of limits of order types by planar measures. Our main result is a rigidity theorem: we show that if sampling two measures induce the same probability distribution on order types, then these measures are projectively equivalent provided the support of at least one of them has non-empty interior. We also show that some condition on the Hausdorff dimension of the support is necessary to obtain projective rigidity and we construct limits of order types that cannot be represented by a planar measure. Returning to combinatorial geometry we relate the regularity of this analytic representation to the aforementioned problem of Erdős on the density of k-tuples in convex position, for large k.

math.CO↗

arXiv2018

Isomorphism of Weighted Trees and Stanley's Conjecture for Caterpillars

This paper contributes to a programme initiated by the first author: `How much information about a graph is revealed in its Potts partition function?'. We show that the $W$-polynomial distinguishes non-isomorphic weighted trees of a \emph{good} family. The framework developed to do so also allows us to show that the $W$-polynomial distinguishes non-isomorphic caterpillars. This establishes Stanley's isomorphism conjecture for caterpillars, an extensively studied problem.

math.CO↗

arXiv2018

Extension from Precoloured Sets of Edges

We consider precolouring extension problems for proper edge-colourings of graphs and multigraphs, in an attempt to prove stronger versions of Vizing's and Shannon's bounds on the chromatic index of (multi)graphs in terms of their maximum degree $Δ$. We are especially interested in the following question: when is it possible to extend a precoloured matching to a colouring of all edges of a (multi)graph? This question turns out to be related to the notorious List Colouring Conjecture and other classic notions of choosability.

math.CO↗

arXiv2017

Do triangle-free planar graphs have exponentially many 3-colorings?

Thomassen conjectured that triangle-free planar graphs have an exponential number of $3$-colorings. We show this conjecture to be equivalent to the following statement: there exists a positive real $α$ such that whenever $G$ is a planar graph and $A$ is a subset of its edges whose deletion makes $G$ triangle-free, there exists a subset $A'$ of $A$ of size at least $α|A|$ such that $G-(A\setminus A')$ is $3$-colorable. This equivalence allows us to study restricted situations, where we can prove the statement to be true.

math.CO↗

arXiv2017

Equitable Colorings of $K\_4$-minor-free Graphs

We demonstrate that for every positive integer $Δ$, every K\_4-minor-free graph with maximum degree $Δ$ admits an equitable coloring with k colors wherek $\ge$ ($Δ$+3)/2. This bound is tight and confirms a conjecture by Zhang and Whu. We do not use the discharging method but rather exploit decomposition trees of K 4-minor-free graphs.

math.CO↗

arXiv2017

Collaborative search on the plane without communication

We generalize the classical cow-path problem [7, 14, 38, 39] into a question that is relevant for collective foraging in animal groups. Specifically, we consider a setting in which k identical (probabilistic) agents, initially placed at some central location, collectively search for a treasure in the two-dimensional plane. The treasure is placed at a target location by an adversary and the goal is to find it as fast as possible as a function of both k and D, where D is the distance between the central location and the target. This is biologically motivated by cooperative, central place foraging such as performed by ants around their nest. In this type of search there is a strong preference to locate nearby food sources before those that are further away. Our focus is on trying to find what can be achieved if communication is limited or altogether absent. Indeed, to avoid overlaps agents must be highly dispersed making communication difficult. Furthermore, if agents do not commence the search in synchrony then even initial communication is problematic. This holds, in particular, with respect to the question of whether the agents can communicate and conclude their total number, k. It turns out that the knowledge of k by the individual agents is crucial for performance. Indeed, it is a straightforward observation that the time required for finding the treasure is $Ω$(D + D 2 /k), and we show in this paper that this bound can be matched if the agents have knowledge of k up to some constant approximation. We present an almost tight bound for the competitive penalty that must be paid, in the running time, if agents have no information about k. Specifically, on the negative side, we show that in such a case, there is no algorithm whose competitiveness is O(log k). On the other hand, we show that for every constant $ε\textgreater{} 0$, there exists a rather simple uniform search algorithm which is $O( \log^{1+ε} k)$-competitive. In addition, we give a lower bound for the setting in which agents are given some estimation of k. As a special case, this lower bound implies that for any constant $ε\textgreater{} 0$, if each agent is given a (one-sided) $k^ε$-approximation to k, then the competitiveness is $Ω$(log k). Informally, our results imply that the agents can potentially perform well without any knowledge of their total number k, however, to further improve, they must be given a relatively good approximation of k. Finally, we propose a uniform algorithm that is both efficient and extremely simple suggesting its relevance for actual biological scenarios.

cs.DC↗