On the density of sets of the Euclidean plane avoiding distance 1

A subset $A \subset \mathbb R^2$ is said to avoid distance $1$ if: $\forall x,y \in A, \left\| x-y \right\|_2 \neq 1.$ In this paper we study the number $m_1(\mathbb R^2)$ which is the supremum of the upper densities of measurable sets avoiding distance 1 in the Euclidean plane. Intuitively, $m_1(\mathbb R^2)$ represents the highest proportion of the plane that can be filled by a set avoiding distance 1. This parameter is related to the fractional chromatic number $\chi_f(\mathbb R^2)$ of the plane. We establish that $m_1(\mathbb R^2) \leq 0.25646$ and $\chi_f(\mathbb R^2) \geq 3.8992$.Comment: 11 pages, 5 figure

On DP-Coloring of Digraphs

DP-coloring is a relatively new coloring concept by Dvo\v{r}\'ak and Postle and was introduced as an extension of list-colorings of (undirected) graphs. It transforms the problem of finding a list-coloring of a given graph $G$ with a list-assignment $L$ to finding an independent transversal in an auxiliary graph with vertex set $\{(v,c) ~|~ v \in V(G), c \in L(v)\}$. In this paper, we extend the definition of DP-colorings to digraphs using the approach from Neumann-Lara where a coloring of a digraph is a coloring of the vertices such that the digraph does not contain any monochromatic directed cycle. Furthermore, we prove a Brooks' type theorem regarding the DP-chromatic number, which extends various results on the (list-)chromatic number of digraphs.Comment: 23 pages, 6 figure

On the density of sets avoiding parallelohedron distance 1

The maximal density of a measurable subset of R^n avoiding Euclidean distance1 is unknown except in the trivial case of dimension 1. In this paper, we consider thecase of a distance associated to a polytope that tiles space, where it is likely that the setsavoiding distance 1 are of maximal density 2^-n, as conjectured by Bachoc and Robins. We prove that this is true for n = 2, and for the Vorono\"i regions of the lattices An, n >= 2

Next Generation Cluster Editing

This work aims at improving the quality of structural variant prediction from the mapped reads of a sequenced genome. We suggest a new model based on cluster editing in weighted graphs and introduce a new heuristic algorithm that allows to solve this problem quickly and with a good approximation on the huge graphs that arise from biological datasets

Homomorphically Full Oriented Graphs

Homomorphically full graphs are those for which every homomorphic image is isomorphic to a subgraph. We extend the definition of homomorphically full to oriented graphs in two different ways. For the first of these, we show that homomorphically full oriented graphs arise as quasi-transitive orientations of homomorphically full graphs. This in turn yields an efficient recognition and construction algorithms for these homomorphically full oriented graphs. For the second one, we show that the related recognition problem is GI-hard, and that the problem of deciding if a graph admits a homomorphically full orientation is NP-complete. In doing so we show the problem of deciding if two given oriented cliques are isomorphic is GI-complete

Separating codes and traffic monitoring

International audienceThis paper studies the problem of traffic monitoring which consists of differentiating a set of walks on a directed graph by placing sensors on as few arcs as possible. The problem of characterising a set of individuals by testing as few attributes as possible is already well-known, but traffic monitoring presents new challenges that the previous models of separation fall short from modelling such as taking into account the multiplicity and order of the arcs in a walk. We introduce a new and stronger model of separation based on languages that generalises the traffic monitoring problem. We study three subproblems with practical applications and develop methods to solve them by combining integer linear programming, separating codes and language theory

Restless Exploration of Periodic Temporal Graphs

A temporal graph is a sequence of graphs, indexed by discrete time steps, with a fixed vertex set but with an edge set that is able to change over time. In the temporal graph exploration problem, an agent wants to visit all the vertices of a given temporal graph. In the classical model, at each time step the agent can either stay where they are, or move along one edge. In this work we add a constraint called restlessness that forces the agent to move along one edge at each time step. We mainly focus on (infinite) periodical temporal graphs. We show that if the period is 2 one can decide in polynomial time whether exploring the whole graph is possible or not, while this problem turns out to be NP-hard for any period p ? 3. We also show some time bounds on the explorations of such graphs when the exploration is possible