Section: New Results
Structural Graph Theory
Participants : JeanClaude Bermond, Frédéric Havet, Nicolas Nisse, Ana Karolinna Maia de Oliveira, Stéphane Pérennes.
More information on several results presented in this section may be found in PhD thesis of A. K. Maia de Oliveira [16] , and in the Habilitation thesis of N. Nisse [17] .
Graph colouring and applications
Graph colouring is a central problem in graph theory and it has a huge number of applications in various scientific domains (telecommunications, scheduling, bioinformatics, ...). We mainly study graph colouring problems that model ressource allocation problems.
Backbone colouring
A wellknown channel assignment problem is the following: we are given a graph $G$, whose vertices correspond to transmitters, together with an edgeweighting $w$. The weight of an edge corresponds to the minimum separation between the channels on its endvertices to avoid interferences. (If there is no edge, no separation is required, the transmitters do not interfere.) We need to assign positive integers (corresponding to channels) to the vertices so that for every edge $e$ the channels assigned to its endvertices differ by at least $w\left(e\right)$. The goal is to minimize the largest integer used, which corresponds to minimizing the span of the used bandwidth. We studied a particular, yet quite general, case, called backbone colouring, in which there are only two levels of interference. So we are given a graph $G$ and a subgraph $H$, called the backone. Two adjacent vertices in $H$ must get integers at least $q$ apart, while adjacent vertices in $G$ must get integers at distance at least 1. The minimum span in this case is called the $q$backbone chromatic number and is denoted $BB{C}_{q}(G,H)$. In [30] and [45] , we focus on the case when $G$ is planar and $H$ is a forest. In [30] , we give a series of NPhardness results as well as upper bounds for $BB{C}_{q}(G,H)$, depending on the type of the forest (matching, galaxy, spanning tree). We also discuss a circular version of the problem. In [45] , we give some upper bounds when $G$ is planar and has no cycles of length 4 an 5, and $G$ is a tree, and we relate those results to the celebrated Steinberg's Conjecture stating that every planar graphs with no cycles of length 4 or 5 is 3colourable.
In [29] , we consider the list version of this problem (in which each vertex is given a particular list of admissible colours), with particular focus on colours in ${\mathcal{Z}}_{p}$ – this problem is closely related to the problem of circular choosability. We first prove that the list circular $q$backbone chromatic number of a graph is bounded by a function of the list chromatic number. We then consider the more general problem in which each edge is assigned an individual distance between its endpoints, and provide bounds using the Combinatorial Nullstellensatz. Through this result and through structural approaches, we achieve good bounds when both the graph and the backbone belong to restricted families of graphs.
Online colouring graphs with few ${P}_{4}$s
Various online colouring procedures are used. The most widespread ones is the greedy one, which results in a greedy colouring. Given a graph $G=(V;E)$, a greedy colouring of $G$ is a proper colouring such that, for each two colours $i<j$, every vertex of $V\left(G\right)$ coloured $j$ has a neighbour with colour $i$. A second optimization procedure consists from time to time to consider the present colouring and to free some colour when possible: if each vertex of a colour class has another colour that is not used by its neighbours, we can recolour each vertex in the calls by another colour. This procedure results in a bcolouring of the graph. A bcolouring of a graph $G$ is a proper colouring such that every colour class contains a vertex which is adjacent to at least one vertex in every other colour class. One of the performance measure of such graph is the maximum numbers of colours they could possibly use. The greatest $k$ such that $G$ has a greedy colouring with $k$ colours is the Grundy number of $G$. The greatest integer $k$ for which there exists a bcolouring of $G$ with $k$ colours is its bchromatic number. Determining the Grundy number and the bchromatic number of a graph are NPhard problems in general. For a fixed $q$, the $(q;q4)$graphs are the graphs for which no set of at most $q$ vertices induces more than $q4$ distinct induced ${P}_{4}$s paths of order 4). In [24] , we obtain polynomialtime algorithms to determine the Grundy number and the bchromatic number of $(q;q4)$graphs, for a fixed $q$. They generalize previous results obtained for cographs and ${P}_{4}$sparse graphs, classes strictly contained in the $(q;q4)$graphs.
Weighted colouring
We also studied weighted colouring which models various problems of shared resources allocation. Given a vertexweighted graph $G$ and a (proper) $r$colouring $c=\{{C}_{1},...,{C}_{r}\}$ of $G$ , the weight of a colour class ${C}_{i}$ is the maximum weight of a vertex coloured $i$ and the weight of $c$ is the sum of the weights of its colour classes. The objective of the Weighted Colouring Problem is, given a vertexweighted graph $G$ , to determine the minimum weight of a proper colouring of $G$, that is, its weighted chromatic number. In [21] , [33] , we prove that the Weighted Coloring Problem admits a version of the Hajós' Theorem and so we show a necessary and sufficient condition for the weighted chromatic number of a vertexweighted graph $G$ to be at least $k$ , for any positive real $k$. The Weighted Colouring Problem problem remains NPcomplete in some particular graph classes as bipartite graphs. In their seminal paper, Guan and Zhu asked whether the weighted chromatic number of bounded treewidth graphs (partial $k$trees) can be computed in polynomialtime. Surprisingly, the timecomplexity of computing this parameter in trees is still open. We show in [21] that, assuming the Exponential Time Hypothesis (3SAT cannot be solved in subexponential time), the best algorithm to compute the weighted chromatic number of $n$node trees has timecomplexity ${n}^{\Theta (logn)}$. Our result mainly relies on proving that, when computing an optimal proper weighted colouring of a graph $G$, it is hard to combine colourings of its connected components, even when $G$ is a forest.
Inducing proper colourings
Frequently, the proper colouring of the graph must be induced by some other parameters that a vertex can compute locally, for example on looking on the labels assigned to its incident edges or to their orientations.
For a connected graph $G$ of order $\leftV\right(G\left)\right\phantom{\rule{0.166667em}{0ex}}\ge \phantom{\rule{0.166667em}{0ex}}3$ and a $k\phantom{\rule{0.166667em}{0ex}}$labelling $c\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}E\left(G\right)\phantom{\rule{0.166667em}{0ex}}\to \phantom{\rule{0.166667em}{0ex}}\{1,2,...,k\}$ of the edges of $G$, the code of a vertex $v$ of $G$ is the ordered $k\phantom{\rule{0.166667em}{0ex}}$tuple $({\ell}_{1},{\ell}_{2},...,{\ell}_{k})$, where ${\ell}_{i}$ is the number of edges incident with $v$ that are labelled $i$. The $k\phantom{\rule{0.166667em}{0ex}}$labelling $c$ is detectable if every two adjacent vertices of $G$ have distinct codes. The minimum positive integer $k$ for which $G$ has a detectable $k\phantom{\rule{0.166667em}{0ex}}$labelling is the detection number $det\left(G\right)$ of $G$. In [31] , we show that it is NPcomplete to decide if the detection number of a cubic graph is 2. We also show that the detection number of every bipartite graph of minimum degree at least 3 is at most 2. Finally, we give some sufficient condition for a cubic graph to have detection number 3.
An orientation of a graph $G$ is a digraph $D$ obtained from $G$ by replacing each edge by exactly one of the two possible arcs with the same endvertices. For each $v\in V\left(G\right)$, the indegree of $v$ in $D$, denoted by ${d}_{D}^{}\left(v\right)$, is the number of arcs with head $v$ in $D$. An orientation $D$ of $G$ is proper if ${d}_{D}^{}\left(u\right)\ne {d}_{D}^{}\left(v\right)$, for all $uv\in E\left(G\right)$. The proper orientation number of a graph $G$, denoted by $po\left(G\right)$, is the minimum of the maximum indegree over all its proper orientations. In [32] , [44] , we prove that $po\left(G\right)\le \left(\Delta \left(G\right)+\sqrt{\Delta \left(G\right)}\right)/2+1$ if $G$ is a bipartite graph, and $po\left(G\right)\le 4$ if $G$ is a tree. It is wellknown that $po\left(G\right)\le \Delta \left(G\right)$, for every graph $G$. However, we prove that deciding whether $po\left(G\right)\le \Delta \left(G\right)1$ is already an $NP$complete problem on graphs with $\Delta \left(G\right)=k$, for every $k\ge 3$. We also show that it is NPcomplete to decide whether $po\left(G\right)\le 2$, for planar subcubic graphs $G$. Moreover, we prove that it is NPcomplete to decide whether $po\left(G\right)\le 3$, for planar bipartite graphs $G$ with maximum degree 5.
Directed graphs
Graph theory can be roughly partitioned into two branches: the areas of undirected graphs and directed graphs (digraphs). Even though both areas have numerous important applications, for various reasons, undirected graphs have been studied much more extensively than directed graphs. One of the reasons is that many problems for digraphs are much more difficult than their analogues for undirected graphs.
Finding a subdivision of a digraph
One of the cornerstones of modern (undirected) graph theory is minor theory of Robertson and Seymour. Unfortunately, we cannot expect an equivalent for directed graphs. Minor theory implies in particular that, for any fixed $F$, detecting a subdivision of $F$ in an input graph $G$ can be performed in polynomial time by the Robertson and Seymour linkage algorithm. In contrast, the analogous subdivision problem for digraph can be either polynomialtime solvable or NPcomplete, depending on the fixed digraph $F$. In [16] , a number of examples of polynomial instances, several NPcompleteness proofs as well as a number of conjectures and open problems are given. In addition, it is conjectured that, for every integer $k$ greater than 1, the directed cycles of length at least $k$ have the ErdősPósa Property : for every $n$, there exists an integer ${t}_{n}$ such that for every digraph $D$, either $D$ contains $n$ disjoint directed cycles of length at least $k$, or there is a set $T$ of ${t}_{n}$ vertices that meets every directed cycle of length at least $k$. This generalizes a celebrated result of Reed, Robertson, Seymour and Thomas which is the case $k=2$ of this conjecture. We prove the conjecture for $k=3$. We also show that the directed $k$Linkage problem is polynomialtime solvable for digraphs with circumference at most 2. From these two results, we deduce that if $F$ is the disjoint union of directed cycles of length at most 3, then one can decide in polynomial time if a digraph contains a subdivision of $F$.
The complexity of finding arcdisjoint branching flows
The concept of arcdisjoint flows in networks is a very general framework within which many wellknown and important problems can be formulated. In particular, the existence of arcdisjoint branching flows, that is, flows which send one unit of flow from a given source $s$ to all other vertices, generalizes the concept of arcdisjoint outbranchings (spanning outtrees) in a digraph. A pair of outbranchings ${B}_{s,1}^{+}$, ${B}_{s,2}^{+}$ from a root s in a digraph $D=(V,A)$ on $n$ vertices corresponds to arcdisjoint branching flows ${x}_{1}$,${x}_{2}$ (the arcs carrying flow in ${x}_{i}$ are those used in ${B}_{s,i}^{+}$, $i=1,2$) in the network that we obtain from $D$ by giving all arcs capacity $n1$. It is then a natural question to ask how much we can lower the capacities on the arcs and still have, say, two arcdisjoint branching flows from the given root $s$. In [46] , we prove that for every fixed integer $\ge 2$ it is

an NPcomplete problem to decide whether a network $\mathcal{N}=(V,A,u)$ where ${u}_{ij}=k$ for every arc $ij$ has two arcdisjoint branching flows rooted at $s$.

a polynomial problem to decide whether a network $\mathcal{N}=(V,A,u)$ on $n$ vertices and ${u}_{ij}=nk$ for every arc $ij$ has two arcdisjoint branching flows rooted at $s$.
The algorithm for the later result generalizes the polynomial algorithm, due to Lovász, for deciding whether a given input digraph has two arcdisjoint outbranchings rooted at a given vertex. Finally we prove that under the socalled Exponential Time Hypothesis (ETH), for every $\u03f5>0$ and for every $k\left(n\right)$ with ${(log\left(n\right))}^{1+\u03f5}\le k\left(n\right)\le \frac{n}{2}$ (and for every large $i$ we have $k\left(n\right)=i$ for some $n$) there is no polynomial algorithm for deciding whether a given digraph contains two arcdisjoint branching flows from the same root so that no arc carries flow larger than $nk\left(n\right)$.
Splitting a tournament into two subtournaments with given minimum outdegree
A $({k}_{1},{k}_{2})$outdegreesplitting of a digraph $D$ is a partition $({V}_{1},{V}_{2})$ of its vertex set such that $D\left[{V}_{1}\right]$ and $D\left[{V}_{2}\right]$ have minimum outdegree at least ${k}_{1}$ and ${k}_{2}$, respectively. In [58] , we show that there exists a minimum function ${f}_{T}$ such that every tournament of minimum outdegree at least ${f}_{T}({k}_{1},{k}_{2})$ has a $({k}_{1},{k}_{2})$outdegreesplitting, and ${f}_{T}({k}_{1},{k}_{2})\le {k}_{1}^{2}/2+3{k}_{1}/2+{k}_{2}+1$. We also show a polynomialtime algorithm that finds a $({k}_{1},{k}_{2})$outdegreesplitting of a tournament if one exists, and returns `no' otherwise. We give better bound on ${f}_{T}$ and faster algorithms when ${k}_{1}=1$.
Eulerian and Hamiltonian dicycles in directed hypergraphs
In [19] , we generalize the concepts of Eulerian and Hamiltonian digraphs to directed hypergraphs. A dihypergraph $H$ is a pair $\left(\mathcal{V}\right(H),\mathcal{E}(H\left)\right)$, where $\mathcal{V}\left(H\right)$ is a nonempty set of elements, called vertices, and $\mathcal{E}\left(H\right)$ is a collection of ordered pairs of subsets of $\mathcal{V}\left(H\right)$, called hyperarcs. It is Eulerian (resp. Hamiltonian) if there is a dicycle containing each hyperarc (resp. each vertex) exactly once. We first present some properties of Eulerian and Hamiltonian dihypergraphs. For example, we show that deciding whether a dihypergraph is Eulerian is an NPcomplete problem. We also study when iterated line dihypergraphs are Eulerian and Hamiltonian. Finally, we study when the generalized de Bruijn dihypergraphs are Eulerian and Hamiltonian. In particular, we determine when they contain a complete Berge dicycle, i.e. an Eulerian and Hamiltonian dicycle.