Section: New Results

Structural Graph Theory

Participants : Jean-Claude 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, bio-informatics, ...). We mainly study graph colouring problems that model ressource allocation problems.

Backbone colouring

A well-known channel assignment problem is the following: we are given a graph $G$, whose vertices correspond to transmitters, together with an edge-weighting $w$. 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 NP-hardness 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 3-colourable.

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 ${𝒵}_p$ – 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.

On-line colouring graphs with few $P_4$s

Various on-line 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 b-colouring of the graph. A b-colouring 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 b-colouring of $G$ with $k$ colours is its b-chromatic number. Determining the Grundy number and the b-chromatic number of a graph are NP-hard problems in general. For a fixed $q$, the $(q;q-4)$-graphs are the graphs for which no set of at most $q$ vertices induces more than $q-4$ distinct induced $P_4$s paths of order 4). In [24] , we obtain polynomial-time algorithms to determine the Grundy number and the b-chromatic number of $(q;q-4)$-graphs, for a fixed $q$. They generalize previous results obtained for cographs and $P_4$-sparse graphs, classes strictly contained in the $(q;q-4)$-graphs.

Weighted colouring

We also studied weighted colouring which models various problems of shared resources allocation. Given a vertex-weighted 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 vertex-weighted 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 vertex-weighted graph $G$ to be at least $k$ , for any positive real $k$. The Weighted Colouring Problem problem remains NP-complete in some particular graph classes as bipartite graphs. In their seminal paper, Guan and Zhu asked whether the weighted chromatic number of bounded tree-width graphs (partial $k$-trees) can be computed in polynomial-time. Surprisingly, the time-complexity of computing this parameter in trees is still open. We show in [21] that, assuming the Exponential Time Hypothesis (3-SAT cannot be solved in sub-exponential time), the best algorithm to compute the weighted chromatic number of $n$-node trees has time-complexity $n^{\Theta (logn)}$. 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 $\left|V\right(G\left)\right|\ge 3$ and a $k$-labelling $c:E\left(G\right)\to \{1,2,...,k\}$ of the edges of $G$, the code of a vertex $v$ of $G$ is the ordered $k$-tuple $({\ell }_1,{\ell }_2,...,{\ell }_k)$, where ${\ell }_i$ is the number of edges incident with $v$ that are labelled $i$. The $k$-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 $k$-labelling is the detection number $det\left(G\right)$ of $G$. In [31] , we show that it is NP-complete 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 well-known 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 NP-complete to decide whether $po\left(G\right)\le 2$, for planar subcubic graphs $G$. Moreover, we prove that it is NP-complete 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 polynomial-time solvable or NP-complete, depending on the fixed digraph $F$. In [16] , a number of examples of polynomial instances, several NP-completeness 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 Erdős-Pósa 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 polynomial-time 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 arc-disjoint branching flows

The concept of arc-disjoint flows in networks is a very general framework within which many well-known and important problems can be formulated. In particular, the existence of arc-disjoint branching flows, that is, flows which send one unit of flow from a given source $s$ to all other vertices, generalizes the concept of arc-disjoint out-branchings (spanning out-trees) in a digraph. A pair of out-branchings $B_{s,1}^{+}$, $B_{s,2}^{+}$ from a root s in a digraph $D=(V,A)$ on $n$ vertices corresponds to arc-disjoint 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 $n-1$. It is then a natural question to ask how much we can lower the capacities on the arcs and still have, say, two arc-disjoint branching flows from the given root $s$. In [46] , we prove that for every fixed integer $\ge 2$ it is

• an NP-complete problem to decide whether a network $𝒩=(V,A,u)$ where $u_{ij}=k$ for every arc $ij$ has two arc-disjoint branching flows rooted at $s$.

• a polynomial problem to decide whether a network $𝒩=(V,A,u)$ on $n$ vertices and $u_{ij}=n-k$ for every arc $ij$ has two arc-disjoint 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 arc-disjoint out-branchings rooted at a given vertex. Finally we prove that under the so-called Exponential Time Hypothesis (ETH), for every $ϵ>0$ and for every $k\left(n\right)$ with ${(log\left(n\right))}^{1+ϵ}\le k\left(n\right)\le \frac{n}{2}$ (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 arc-disjoint branching flows from the same root so that no arc carries flow larger than $n-k\left(n\right)$.

Splitting a tournament into two subtournaments with given minimum outdegree

A $(k_1,k_2)$-outdegree-splitting 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 $(k_1,k_2)$-outdegree-splitting, and $f_T(k_1,k_2)\le k_1^2/2+3k_1/2+k_2+1$. We also show a polynomial-time algorithm that finds a $(k_1,k_2)$-outdegree-splitting 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 $\left(𝒱\right(H),ℰ(H\left)\right)$, where $𝒱\left(H\right)$ is a non-empty set of elements, called vertices, and $ℰ\left(H\right)$ is a collection of ordered pairs of subsets of $𝒱\left(H\right)$, 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 NP-complete 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.