PICASARRI-ARRIETA , Lucas

Let $D=(V,A)$ be a digraph. We define $\Delta_{\max}(D)$ as the maximum of $\{ \max(d^+(v),d^-(v)) \mid v \in V \}$ and $\Delta_{\min}(D)$ as the maximum of $\{ \min(d^+(v),d^-(v)) \mid v \in V \}$. It is known that the dichromatic number of $D$ is at most $\Delta_{\min}(D) + 1$. In this work, we prove that every digraph $D$ which has...

Let D = (V, A) be a digraph. We define ∆max(D) as the maximum of {max(d + (v), d-(v)) | v ∈ V } and ∆min(D) as the maximum of {min(d + (v), d-(v)) | v ∈ V }. It is known that the dichromatic number of D is at most ∆min(D) + 1. In this work, we prove that every digraph D which has dichromatic number exactly ∆min(D) + 1 must contain the directed...

This thesis focuses on a notion of colouring of digraphs introduced by ErdH{o}s and Neumann-Lara in the late 1970s, namely the dicolouring, and its associated digraph parameter: the dichromatic number. It appears in the last decades that many classical results on proper graph colouring have directed counterparts using these notions.We first...

The dichromatic number $\vec{\chi}(D)$ of a digraph $D$ is the least integer $k$ for which $D$ has a coloring with $k$ colors such that there is no monochromatic directed cycle in $D$. The digraphs considered here are finite and may have antiparallel arcs, but no parallel arcs. A digraph $D$ is called $k$-critical if each proper subdigraph $D'$...

We consider two decomposition problems in directed graphs. We say that a digraph is $k$-bounded for some $k \in \mathbb{Z}_{\geq 1}$ if each of its connected components contains at most $k$ arcs. For the first problem, a directed linear forest is a collection of vertex-disjoint directed paths and we consider the problem of decomposing a given...

Aboulker et al. proved that a digraph with large enough dichromatic number contains any fixed digraph as a subdivision. The dichromatic number of a digraph is the smallest order of a partition of its vertex set into acyclic induced subdigraphs. A digraph is dicritical if the removal of any arc or vertex decreases its dichromatic number. In this...

Reed in 1998 conjectured that every graph $G$ satisfies $\chi(G) \leq \lceil \frac{\Delta(G)+1+\omega(G)}{2} \rceil$. As a partial result, he proved the existence of $\varepsilon > 0$ for which every graph $G$ satisfies $\chi(G) \leq \lceil (1-\varepsilon)(\Delta(G)+1)+\varepsilon\omega(G) \rceil$. We propose an analogue conjecture for...

Given a digraph D = (V, A) on n vertices and a vertex v ∈ V , the cycle-degree of v is the minimum size of a set S ⊆ V (D) \ {v} intersecting every directed cycle of D containing v. From this definition of cycle-degree, we define the c-degeneracy (or cycle-degeneracy) of D, which we denote by δ * c (D). It appears to be a nice generalisation of...

The dichromatic number of a digraph is the minimum integer $k$ such that it admits a $k$-dicolouring, i.e. a partition of its vertices into $k$ acyclic subdigraphs. We say that a digraph $D$ is a super-orientation of an undirected graph $G$ if $G$ is the underlying graph of $D$. If $D$ does not contain any pair of symmetric arcs, we just say...

We prove that every 4-dicritical oriented graph on n vertices has at least $(\frac{10}{3}+\frac{1}{51})n - 1$ arcs.

A digraph is $3$-dicritical if it cannot be vertex-partitioned into two sets inducing acyclic digraphs, but each of its proper subdigraphs can. We give a human-readable proof that the number of 3-dicritical semi-complete digraphs is finite. Further, we give a computer-assisted proof of a full characterization of 3-dicritical semi-complete...

The dichromatic number of a digraph is the minimum number of colours needed to colour the vertices of a digraph such that each colour class induces an acyclic subdigraph. A digraph is ‐dicritical if and each proper subdigraph of satisfies . For integers and , we define (resp., ) as the minimum number of arcs possible in a ‐dicritical digraph...

Brooks' Theorem is a fundamental result on graph colouring, stating that the chromatic number of a graph is almost always upper bounded by its maximal degree. Lov\'asz showed that such a colouring may then be computed in linear time when it exists. Many analogues are known for variants of (di)graph colouring, notably for list-colouring and...

The support of a flow $x$ in a network is the subdigraph induced by the arcs $ij$ for which $x_{ij}>0$. We discuss a number of results on flows in networks where we put certain restrictions on structure of the support of the flow. Many of these problems are NP-hard because they generalize linkage problems for digraphs. For example deciding...

Let $K$ be a complete graph of order $n$. For $d\in (0,1)$, let $c$ be a $\pm 1$-edge labeling of $K$ such that there are $d{n\choose 2}$ edges with label $+1$, and let $G$ be a spanning subgraph of $K$ of maximum degree at most $\Delta$ and with $m(G)$ edges. We prove the existence of an isomorphic copy $G'$ of $G$ in $K$ such that the number...

A classical result of Erdős and Gallai determines the maximum size $m(n,\nu)$ of a graph $G$ of order $n$ and matching number $\nu n$. We show that $G$ has factorially many maximum matchings provided that its size is sufficiently close to $m(n,\nu)$.

We study the complexity of deciding whether a given digraph D = (V, A) admits a partition (A1, A2) of its arc set such that each of the corresponding digraphs D1 = (V, A1) and D2 = (V, A2) satisfy some given prescribed property. We mainly focus on the following 15 properties: being bipartite, being connected, being strongly connected, being...

In this work, we generalize several results on graph recolouring to digraphs. Given two k-dicolourings of a digraph D, we prove that it is PSPACE-complete to decide whether we can transform one into the other by recolouring one vertex at each step while maintaining a dicolouring at any step even for k = 2 and for digraphs with maximum degree 5...

In this work, we generalize several results on graph recolouring to digraphs. Given two k-dicolourings of a digraph D, we prove that it is PSPACE-complete to decide whether we can transform one into the other by recolouring one vertex at each step while maintaining a dicolouring at any step even for k = 2 and for digraphs with maximum degree 5...