Pierre Charbit
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View article: Extending Ghouila-Houri's Characterization of Comparability Graphs to Temporal Graphs
Extending Ghouila-Houri's Characterization of Comparability Graphs to Temporal Graphs Open
An orientation of a given static graph is called transitive if for any three vertices $a,b,c$, the presence of arcs $(a,b)$ and $(b,c)$ forces the presence of the arc $(a,c)$. If only the presence of an arc between $a$ and $c$ is required,…
View article: ($\overrightarrow{P_6}$, triangle)-Free Digraphs Have Bounded Dichromatic Number
($\overrightarrow{P_6}$, triangle)-Free Digraphs Have Bounded Dichromatic Number Open
The dichromatic number of an oriented graph is the minimum size of a partition of its vertices into acyclic induced subdigraphs. We prove that oriented graphs with no induced directed path on six vertices and no triangle have bounded dichr…
View article: Dichromatic Number and Cycle Inversions
Dichromatic Number and Cycle Inversions Open
The results of this note were stated in the first author PhD manuscript in 2006 but never published. The writing of a proof given there was slightly careless and the proof itself scattered across the document, the goal of this note is to g…
View article: On the Limits of Information Spread by Memory-Less Agents
On the Limits of Information Spread by Memory-Less Agents Open
We address the self-stabilizing bit-dissemination problem, designed to capture the challenges of spreading information and reaching consensus among entities with minimal cognitive and communication capacities. Specifically, a group of n ag…
View article: Improved Pyrotechnics: Closer to the Burning Number Conjecture
Improved Pyrotechnics: Closer to the Burning Number Conjecture Open
The Burning Number Conjecture claims that for every connected graph $G$ of order $n,$ its burning number satisfies $b(G) \le \lceil \sqrt{n}\, \rceil.$ While the conjecture remains open, we prove that it is asymptotically true when the ord…
View article: Clique number of tournaments
Clique number of tournaments Open
We introduce the notion of clique number of a tournament and investigate its relation with the dichromatic number. In particular, it permits defining $\dic$-bounded classes of tournaments, which is the paper's main topic.
View article: Digraph Colouring and Arc-Connectivity
Digraph Colouring and Arc-Connectivity Open
The dichromatic number $\vecχ(D)$ of a digraph $D$ is the minimum size of a partition of its vertices into acyclic induced subgraphs. We denote by $λ(D)$ the maximum local edge connectivity of a digraph $D$. Neumann-Lara proved that for ev…
View article: ( #» P 6 , triangle)-free digraphs have bounded dichromatic number
( #» P 6 , triangle)-free digraphs have bounded dichromatic number Open
The dichromatic number of an oriented graph is the minimum size of a partition of its vertices into acyclic induced subdigraphs. We prove that oriented graphs with no induced directed path on six vertices and no triangle have bounded dichr…
View article: (P6, triangle)-free digraphs have bounded dichromatic number
(P6, triangle)-free digraphs have bounded dichromatic number Open
The dichromatic number of an oriented graph is the minimum size of a partition of its vertices into acyclic induced subdigraphs. We prove that oriented graphs with no induced directed path on six vertices and no triangle have bounded dichr…
View article: Heroes in oriented complete multipartite graphs
Heroes in oriented complete multipartite graphs Open
The dichromatic number of a digraph is the minimum size of a partition of its vertices into acyclic induced subgraphs. Given a class of digraphs $\mathcal C$, a digraph $H$ is a hero in $\mc C$ if $H$-free digraphs of $\mathcal C$ have bou…
View article: Edge clique covers in graphs with independence number two
Edge clique covers in graphs with independence number two Open
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View article: Improved pyrotechnics : Closer to the burning graph conjecture
Improved pyrotechnics : Closer to the burning graph conjecture Open
The Burning Number Conjecture claims that for every connected graph $G$ of order $n,$ its burning number satisfies $b(G) \le \lceil \sqrt{n} \rceil.$ While the conjecture remains open, we prove that it is asymptotically true when the order…
View article: Extension of Gyárfás-Sumner Conjecture to Digraphs
Extension of Gyárfás-Sumner Conjecture to Digraphs Open
The dichromatic number of a digraph $D$ is the minimum number of colors needed to color its vertices in such a way that each color class induces an acyclic digraph. As it generalizes the notion of the chromatic number of graphs, it has bec…
View article: Abstracting Linear Equation Systems
Abstracting Linear Equation Systems Open
We study the problem of how to convert reaction networks into boolean networks. We start from the sign abstraction of the ODE semantics of reaction network, and show that it can be captured by a quasi-boolean network, a generalization of b…
View article: Parameterized Complexity of Independent Set in H-Free Graphs
Parameterized Complexity of Independent Set in H-Free Graphs Open
In this paper, we investigate the complexity of Maximum Independent Set (MIS) in the class of H-free graphs, that is, graphs excluding a fixed graph as an induced subgraph. Given that the problem remains NP-hard for most graphs H, we study…
View article: Deriving Proved Equality Tests in Coq-Elpi: Stronger Induction Principles for Containers in Coq
Deriving Proved Equality Tests in Coq-Elpi: Stronger Induction Principles for Containers in Coq Open
We describe a procedure to derive equality tests and their correctness proofs from inductive type declarations in Coq. Programs and proofs are derived compositionally, reusing code and proofs derived previously. The key steps are two. Firs…
View article: Enclosings of Decompositions of Complete Multigraphs in $2$-Edge-Connected $r$-Factorizations
Enclosings of Decompositions of Complete Multigraphs in $2$-Edge-Connected $r$-Factorizations Open
A decomposition of a multigraph $G$ is a partition of its edges into subgraphs $G(1), \ldots , G(k)$. It is called an $r$-factorization if every $G(i)$ is $r$-regular and spanning. If $G$ is a subgraph of $H$, a decomposition of $G$ is sai…
View article: Parameterized Complexity of Independent Set in H-Free Graphs
Parameterized Complexity of Independent Set in H-Free Graphs Open
In this paper, we investigate the complexity of Maximum Independent Set (MIS) in the class of $H$-free graphs, that is, graphs excluding a fixed graph as an induced subgraph. Given that the problem remains $NP$-hard for most graphs $H$, we…
View article: EPTAS for Max Clique on Disks and Unit Balls
EPTAS for Max Clique on Disks and Unit Balls Open
We propose a polynomial-time algorithm which takes as input a finite set of points of $\mathbb R^3$ and compute, up to arbitrary precision, a maximum subset with diameter at most $1$. More precisely, we give the first randomized EPTAS and …
View article: Issue Information
Issue Information Open
The 2-surviving rate of planar graphs with average degree lower than 9 -
View article: χ‐bounded families of oriented graphs
χ‐bounded families of oriented graphs Open
A famous conjecture of Gyárfás and Sumner states for any tree T and integer k , if the chromatic number of a graph is large enough, either the graph contains a clique of size k or it contains T as an induced subgraph. We discuss some resul…
View article: EPTAS for Max Clique on Disks and Unit Balls
EPTAS for Max Clique on Disks and Unit Balls Open
We propose a polynomial-time algorithm which takes as input a finite set of points of $\mathbb R^3$ and compute, up to arbitrary precision, a maximum subset with diameter at most $1$. More precisely, we give the first randomized EPTAS and …
View article: The Salesman's Improved Tours for Fundamental Classes
The Salesman's Improved Tours for Fundamental Classes Open
Finding the exact integrality gap $α$ for the LP relaxation of the metric Travelling Salesman Problem (TSP) has been an open problem for over thirty years, with little progress made. It is known that $4/3 \leq α\leq 3/2$, and a famous conj…
View article: Layers and Matroids for the Traveling Salesman's Paths
Layers and Matroids for the Traveling Salesman's Paths Open
Gottschalk and Vygen proved that every solution of the subtour elimination linear program for traveling salesman paths is a convex combination of more and more restrictive "generalized Gao-trees". We give a short proof of this fact, as a l…
View article: A New Graph Parameter To Measure Linearity
A New Graph Parameter To Measure Linearity Open
Consider a sequence of LexBFS vertex orderings σ1, σ2, . . . where each ordering σi is used to break ties for σi+1. Since the total number of vertex orderings of a finite graph is finite, this sequence must end in a cycle of vertex orderin…