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François Pirot

Publications and source records attributed to François Pirot.

At least 19 recordsLinked to original sources
arXiv2026

A Domatic Analogue of $χ$-Bounded Graph Classes and the Gyárfás-Sumner Conjecture

Given a graph $G$, a dominating set is a subset $X\subseteq V(G)$ such that $N[X]=V(G)$. The \emph{domatic number} of $G$, denoted ${\rm dom}(G)$, is the maximum size of a partition of $V(G)$ into dominating sets. In analogy with the lower bound of the chromatic number by the clique number, the domatic number satisfies the upper bound ${\rm dom}(G)\le δ(G)+1$ where $δ(G)$ is the minimum degree of $G$. Therefore, as an analogue of the notion of $χ$-bounded graph classes, we say that a class of graphs $\mathscr{G}$ is \emph{DOM-bounded} if there exists a positive unbounded function $f_{\mathscr{G}}$ such that for every $G\in \mathscr{G}$, we have ${\rm dom}(G) \ge f_{\mathscr{G}}(δ(G))$. We propose the following conjecture for graphs forbidding a fixed induced subgraph, analogous to the Gyárfás--Sumner Conjecture for $χ$-bounded graph classes: for every connected graph $H$, the class of $H$-free graphs is DOM-bounded if and only if $H$ is a tree of diameter at most $3$. We reduce the case of disconnected graphs to the connected setting and show that the conditions on $H$ are necessary. We show that star-free graphs of minimum degree at least $δ$ have domatic number $Ω(δ/\log δ)$, which is best possible up to a constant factor. We also identify a subclass of star-free graphs for which the domatic number is linear in $δ$: line graphs of bounded rank hypergraphs. In support of our conjecture in the case of double stars, we prove that $P_4$-free graphs (i.e. cographs) of minimum degree $δ$ have domatic number at least $1 + \fracδ{2}$, which is best possible.

math.CO↗
arXiv2026

Colouring the interference digraph of a set of requests in a bidirected tree

In this paper, we investigate the impact of the broadcast effect arising in filterless optical networks on the computational complexity of the wavelength assignment problem. We model conflicts using an appropriate interference digraph, whose proper colourings correspond to feasible wavelength assignments. Minimizing the number of required wavelengths therefore amounts to determining the chromatic number of this interference digraph. Within this framework, we first present a polynomial-time 2-approximation algorithm for minimizing the number of wavelengths. We then show that the problem is fixed-parameter tractable when parameterized by the number $k$ of available wavelengths. We also derive polynomial-time algorithms for computing the independence and clique numbers of this interference digraph.

cs.DM↗
arXiv2026

New bounds for proper $h$-conflict-free colourings

A proper $k$-colouring of a graph $G$ is called $h$-conflict-free if every vertex $v$ has at least $\min\, \{h, {\rm deg}(v)\}$ colours appearing exactly once in its neighbourhood. Let $χ_{\rm pcf}^h(G)$ denote the minimum $k$ such that such a colouring exists. We show that for every fixed $h\ge 1$, every graph $G$ of maximum degree $Δ$ satisfies $χ_{\rm pcf}^h(G) \le hΔ+ \mathcal{O}(\log Δ)$. This expands on the work of Cho et al., and improves a recent result of Liu and Reed in the case $h=1$. We conjecture that for every $h\ge 1$ and every graph $G$ of maximum degree $Δ$ sufficiently large, the bound $χ_{\rm pcf}^h(G) \le hΔ+ 1$ should hold, which would be tight. When the minimum degree $δ$ of $G$ is sufficiently large, namely $δ\ge \max\{100h, 2000\log Δ\}$, we show that this upper bound can be further reduced to $χ_{\rm{pcf}}^h(G) \le Δ+ \mathcal{O}(\sqrt{hΔ})$. This improves a recent bound from Kamyczura and Przybyło when $δ\le \sqrt{hΔ}$.

math.CO↗
arXiv2026

Acyclic colourings of graphs with obstructions

Given a graph $G$, a colouring of $G$ is \emph{acyclic} if it is a proper colouring of $G$ and every cycle contains at least three colours. Its acyclic chromatic number $χ_a(G)$ is the minimum~$k$ such that an acyclic $k$-colouring of $G$ exists. When $G$ has maximum degree $Δ$, it is known that $χ_a(G) = \mathcal {O}(Δ^{4/3})$ as $Δ\to \infty$, and that $χ_a(G) = \mathcal {O}(\sqrt{t} \cdot Δ)$ if in addition $G$ does not contain $K_{2,t}$ as a subgraph. We study the extremal value of the acyclic chromatic number in the class of graphs of maximum degree $Δ$ that do not contain some fixed subgraph $F$ on $t$ vertices. We establish that this extremal value is at most $\mathcal {O}(t^{8/3}Δ^{2/3})$ if $F$ is a tree, $\mathcal {O}(\sqrt{t} \cdot Δ)$ if $F$ is bipartite and can be made acyclic with the removal of one vertex, $2Δ+ \mathcal {O}(tΔ^{2/3})$ if $F$ is an even cycle of length at least $6$, and $\mathcal {O}(t^{1/4}Δ^{5/4})$ if $F=K_{3,t}$. Moreover, we exhibit an infinite family of obstructions $F$ that each induces a different asymptotic behaviour for this extremal value. This is obtained with the derivation of lower bounds that come from the analysis of the acyclic chromatic number of a random graph drawn from either $G(n,p)$ or $G(n,n,p)$, that we entirely determine up to a ${\rm polylog}(n)$ factor. As a byproduct, we can certify that most of our results are tight up to a $Δ^{\mathcal{O}(1/t)}$ factor.

math.CO↗
arXiv2025

Fractional domatic number and minimum degree

The domatic number of a graph $G$ is the maximum number of pairwise disjoint dominating sets of $G$. We are interested in the LP-relaxation of this parameter, which is called the fractional domatic number of $G$. We study its extremal value in the class of graphs of minimum degree $d$. The fractional domatic number of a graph of minimum degree $d$ is always at most $d+1$, and at least $(1-o(1))\, d/\ln d$ as $d\to \infty$. This is asymptotically tight even within the class of split graphs. Our main result concerns the case $d=2$; we show that, excluding $8$ exceptional graphs, the fractional domatic number of every connected graph of minimum degree (at least) $2$ is at least $5/2$. We also show that this bound cannot be improved if only finitely many graphs are excluded, even when restricting to bipartite graphs of girth at least $6$. This proves in a stronger sense a conjecture by Gadouleau, Harms, Mertzios, and Zamaraev (2024). This also extends and generalises results from McCuaig and Shepherd (1989), from Fujita, Kameda, and Yamashita (2000), and from Abbas, Egerstedt, Liu, Thomas, and Whalen (2016). Finally, we show that planar graphs of minimum degree at least $2$ and girth at least $g$ have fractional domatic number at least $3 - O(1/g)$ as $g\to\infty$.

math.CO↗
arXiv2025

Chromatic discrepancy of locally $s$-colourable graphs

The chromatic discrepancy of a graph $G$, denoted $ϕ(G)$, is the least over all proper colourings $σ$ of $G$ of the greatest difference between the number of colours $|σ(V(H))|$ spanned by an induced subgraph $H$ of $G$ and its chromatic number $χ(H)$. We prove that the chromatic discrepancy of a triangle-free graph $G$ is at least $χ(G)-2$. This is best possible and positively answers a question raised by Aravind, Kalyanasundaram, Sandeep, and Sivadasan. More generally, we say that a graph $G$ is locally $s$-colourable if the closed neighbourhood of any vertex $v\in V(G)$ is properly $s$-colourable; in particular, a triangle-free graph is locally $2$-colourable. We conjecture that every locally $s$-colourable graph $G$ satisfies $ϕ(G) \geq χ(G)-s$, and show that this would be almost best possible. We prove the conjecture when $χ(G) \le 11s/6$, and as a partial result towards the general case, we prove that every locally $s$-colourable graph $G$ satisfies $ϕ(G) \geq χ(G) - s\ln χ(G)$. If the conjecture holds, it implies in particular, for every integer $\ell\geq 2$, that any graph $G$ without any copy of $C_{\ell+1}$, the cycle of length $\ell+1$, satisfies $ϕ(G) \geq χ(G) - \ell$. When $\ell \ge 3$ and $G\neq K_\ell$, we conjecture that we actually have $ϕ(G)\ge χ(G) - \ell + 1$, and prove it in the special case $\ell = 3$ or $χ(G) \le 5\ell/3$. In general, we further obtain that every $C_{\ell+1}$-free graph $G$ satisfies $ϕ(G) \geq χ(G) - O_{\ell}(\ln \ln χ(G))$. We do so by determining an almost tight bound on the chromatic number of balls of radius at most $\ell/2$ in $G$, which could be of independent interest.

math.CO↗
arXiv2024

Self-stabilization and byzantine tolerance for maximal independent

We analyze the impact of transient and Byzantine faults on the construction of a maximal independent set in a general network. We adapt the self-stabilizing algorithm presented by Turau `for computing such a vertex set. Our algorithm is self-stabilizing and also works under the more difficult context of arbitrary Byzantine faults. Byzantine nodes can prevent nodes close to them from taking part in the independent set for an arbitrarily long time. We give boundaries to their impact by focusing on the set of all nodes excluding nodes at distance 1 or less of Byzantine nodes, and excluding some of the nodes at distance 2. As far as we know, we present the first algorithm tolerating both transient and Byzantine faults under the fair distributed daemon. We prove that this algorithm converges in $ \mathcal O(Δn)$ rounds w.h.p., where $n$ and $Δ$ are the size and the maximum degree of the network, resp. Additionally, we present a modified version of this algorithm for anonymous systems under the adversarial distributed daemon that converges in $ \mathcal O(n^{2})$ expected number of steps.

cs.DC↗
arXiv2024

Minimum acyclic number and maximum dichromatic number of oriented triangle-free graphs of a given order

Let $D$ be a digraph. Its acyclic number $\vecα(D)$ is the maximum order of an acyclic induced subdigraph and its dichromatic number $\vecχ(D)$ is the least integer $k$ such that $V(D)$ can be partitioned into $k$ subsets inducing acyclic subdigraphs. We study ${\vec a}(n)$ and $\vec t(n)$ which are the minimum of $\vecα(D)$ and the maximum of $\vecχ(D)$, respectively, over all oriented triangle-free graphs of order $n$. For every $ε>0$ and $n$ large enough, we show $(1/\sqrt{2} - ε) \sqrt{n\log n} \leq \vec{a}(n) \leq \frac{107}{8} \sqrt n \log n$ and $\frac{8}{107} \sqrt n/\log n \leq \vec{t}(n) \leq (\sqrt 2 + ε) \sqrt{n/\log n}$. We also construct an oriented triangle-free graph on 25 vertices with dichromatic number~3, and show that every oriented triangle-free graph of order at most 17 has dichromatic number at most 2.

math.CO↗
arXiv2023

New bounds for odd colourings of graphs

Given a graph $G$, a vertex-colouring $σ$ of $G$, and a subset $X\subseteq V(G)$, a colour $x \in σ(X)$ is said to be \emph{odd} for $X$ in $σ$ if it has an odd number of occurrences in $X$. We say that $σ$ is an \emph{odd colouring} of $G$ if it is proper and every (open) neighbourhood has an odd colour in $σ$. The odd chromatic number of a graph $G$, denoted by $χ_o(G)$, is the minimum $k\in\mathbb{N}$ such that an odd colouring $σ\colon V(G)\to [k]$ exists. In a recent paper, Caro, Petru\v sevski and \v Skrekovski conjectured that every connected graph of maximum degree $Δ\ge 3$ has odd-chromatic number at most $Δ+1$. We prove that this conjecture holds asymptotically: for every connected graph $G$ with maximum degree $Δ$, $χ_o(G)\leΔ+O(\lnΔ)$ as $Δ\to \infty$. We also prove that $χ_o(G)\le\lfloor3Δ/2\rfloor+2$ for every $Δ$. If moreover the minimum degree $δ$ of $G$ is sufficiently large, we have $χ_o(G) \le χ(G) + O(Δ\ln Δ/δ)$ and $χ_o(G) = O(χ(G)\ln Δ)$. Finally, given an integer $h\ge 1$, we study the generalisation of these results to $h$-odd colourings, where every vertex $v$ must have at least $\min \{°(v),h\}$ odd colours in its neighbourhood. Many of our results are tight up to some multiplicative constant.

math.CO↗
arXiv2023

Uniformly Random Colourings of Sparse Graphs

We analyse uniformly random proper $k$-colourings of sparse graphs with maximum degree $Δ$ in the regime $Δ< k\ln k $. This regime corresponds to the lower side of the shattering threshold for random graph colouring, a paradigmatic example of the shattering threshold for random Constraint Satisfaction Problems. We prove a variety of results about the solution space geometry of colourings of fixed graphs, generalising work of Achlioptas, Coja-Oghlan, and Molloy on random graphs, and justifying the performance of stochastic local search algorithms in this regime. Our central proof relies only on elementary techniques, namely the first-moment method and a quantitative induction, yet it strengthens list-colouring results due to Vu, and more recently Davies, Kang, P., and Sereni, and generalises state-of-the-art bounds from Ramsey theory in the context of sparse graphs. It further yields an approximately tight lower bound on the number of colourings, also known as the partition function of the Potts model, with implications for efficient approximate counting.

math.CO↗
arXiv2021

Colouring locally sparse graphs with the first moment method

We give a short proof of a bound on the list chromatic number of graphs $G$ of maximum degree $Δ$ where each neighbourhood has density at most $d$, namely $χ_\ell(G) \le (1+o(1)) \fracΔ{\ln \fracΔ{d+1}}$ as $\fracΔ{d+1} \to \infty$. This bound is tight up to an asymptotic factor $2$, which is the best possible barring a breakthrough in Ramsey theory, and strengthens results due to Vu, and more recently Davies, P., Kang, and Sereni. Our proof relies on the first moment method, and adapts a clever counting argument developed by Rosenfeld in the context of non-repetitive colourings. As a final touch, we show that our method provides an asymptotically tight lower bound on the number of colourings of locally sparse graphs.

math.CO↗
arXiv2021

Strong chromatic index and Hadwiger number

We investigate the effect of a fixed forbidden clique minor upon the strong chromatic index, both in multigraphs and in simple graphs. We conjecture for each $k\ge 4$ that any $K_k$-minor-free multigraph of maximum degree $Δ$ has strong chromatic index at most $\frac32(k-2)Δ$. We present a construction certifying that if true the conjecture is asymptotically sharp as $Δ\to\infty$. In support of the conjecture, we show it in the case $k=4$ and prove the statement for strong clique number in place of strong chromatic index. By contrast, we make a basic observation that for $K_k$-minor-free simple graphs, the problem of strong edge-colouring is "between" Hadwiger's Conjecture and its fractional relaxation. For $k\geq5$, we also show that $K_k$-minor-free multigraphs of edge-diameter at most $2$ have strong clique number at most $(k-\frac{1}{2})Δ$.

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↗
arXiv2021

Asymptotic Dimension of Minor-Closed Families and Assouad-Nagata Dimension of Surfaces

The asymptotic dimension is an invariant of metric spaces introduced by Gromov in the context of geometric group theory. In this paper, we study the asymptotic dimension of metric spaces generated by graphs and their shortest path metric and show their applications to some continuous spaces. The asymptotic dimension of such graph metrics can be seen as a large scale generalisation of weak diameter network decomposition which has been extensively studied in computer science. We prove that every proper minor-closed family of graphs has asymptotic dimension at most 2, which gives optimal answers to a question of Fujiwara and Papasoglu and (in a strong form) to a problem raised by Ostrovskii and Rosenthal on minor excluded groups. For some special minor-closed families, such as the class of graphs embeddable in a surface of bounded Euler genus, we prove a stronger result and apply this to show that complete Riemannian surfaces have Assouad-Nagata dimension at most 2. Furthermore, our techniques allow us to prove optimal results for the asymptotic dimension of graphs of bounded layered treewidth and graphs of polynomial growth, which are graph classes that are defined by purely combinatorial notions and properly contain graph classes with some natural topological and geometric flavours.

math.CO↗
arXiv2021

Distributed algorithms for fractional coloring

In this paper we study fractional coloring from the angle of distributed computing. Fractional coloring is the linear relaxation of the classical notion of coloring, and has many applications, in particular in scheduling. It was proved by Hasemann, Hirvonen, Rybicki and Suomela (2016) that for every real $α>1$ and integer $Δ$, a fractional coloring of total weight at most $α(Δ+1)$ can be obtained deterministically in a single round in graphs of maximum degree $Δ$, in the LOCAL model of computation. However, a major issue of this result is that the output of each vertex has unbounded size. Here we prove that even if we impose the more realistic assumption that the output of each vertex has constant size, we can find fractional colorings of total weight arbitrarily close to known tight bounds for the fractional chromatic number in several cases of interest. More precisely, we show that for any fixed $ε> 0$ and $Δ$, a fractional coloring of total weight at most $Δ+ε$ can be found in $O(\log^*n)$ rounds in graphs of maximum degree $Δ$ with no $K_{Δ+1}$, while finding a fractional coloring of total weight at most $Δ$ in this case requires $Ω(\log \log n)$ rounds for randomized algorithms and $Ω( \log n)$ rounds for deterministic algorithms. We also show how to obtain fractional colorings of total weight at most $2+ε$ in grids of any fixed dimension, for any $ε>0$, in $O(\log^*n)$ rounds. Finally, we prove that in sparse graphs of large girth from any proper minor-closed family we can find a fractional coloring of total weight at most $2+ε$, for any $ε>0$, in $O(\log n)$ rounds.

cs.DC↗
arXiv2020

Occupancy fraction, fractional colouring, and triangle fraction

Given $\varepsilon>0$, there exists $f_0$ such that, if $f_0 \le f \le Δ^2+1$, then for any graph $G$ on $n$ vertices of maximum degree $Δ$ in which the neighbourhood of every vertex in $G$ spans at most $Δ^2/f$ edges, (i) an independent set of $G$ drawn uniformly at random has at least $(1/2-\varepsilon)(n/Δ)\log f$ vertices in expectation, and (ii) the fractional chromatic number of $G$ is at most $(2+\varepsilon)Δ/\log f$. These bounds cannot in general be improved by more than a factor $2$ asymptotically. One may view these as stronger versions of results of Ajtai, Komlós and Szemerédi (1981) and Shearer (1983). The proofs use a tight analysis of the hard-core model.

math.CO↗
arXiv2020

Surfaces have (asymptotic) dimension 2

The asymptotic dimension is an invariant of metric spaces introduced by Gromov in the context of geometric group theory. When restricted to graphs and their shortest paths metric, the asymptotic dimension can be seen as a large scale version of weak diameter colorings (also known as weak diameter network decompositions), i.e. colorings in which each monochromatic component has small weak diameter. In this paper, we prove that for any $p$, the class of graphs excluding $K_{3,p}$ as a minor has asymptotic dimension at most 2. This implies that the class of all graphs embeddable on any fixed surface (and in particular the class of planar graphs) has asymptotic dimension 2, which gives a positive answer to a recent question of Fujiwara and Papasoglu. Our result extends from graphs to Riemannian surfaces. We also prove that graphs of bounded pathwidth have asymptotic dimension at most 1 and graphs of bounded layered pathwidth have asymptotic dimension at most 2. We give some applications of our techniques to graph classes defined in a topological or geometrical way, and to graph classes of polynomial growth. Finally we prove that the class of bounded degree graphs from any fixed proper minor-closed class has asymptotic dimension at most 2. This can be seen as a large scale generalization of the result that bounded degree graphs from any fixed proper minor-closed class are 3-colorable with monochromatic components of bounded size. This also implies that (infinite) Cayley graphs avoiding some minor have asymptotic dimension at most 2, which solves a problem raised by Ostrovskii and Rosenthal.

math.CO↗
arXiv2020

Packing and covering balls in graphs excluding a minor

We prove that for every integer $t\ge 1$ there exists a constant $c_t$ such that for every $K_t$-minor-free graph $G$, and every set $S$ of balls in $G$, the minimum size of a set of vertices of $G$ intersecting all the balls of $S$ is at most $c_t$ times the maximum number of vertex-disjoint balls in $S$. This was conjectured by Chepoi, Estellon, and Vaxès in 2007 in the special case of planar graphs and of balls having the same radius.

math.CO↗