Section: New Results

Structural Graph Theory

Participants : Julio Araújo, Jean-Claude Bermond, Frédéric Havet, Nicolas Nisse, Ana Karolinna Maia de Oliveira, Stéphane Pérennes.

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-weghting $w$. 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 is this case is called the $q$-backbone chromatic number and is denoted $BB{C}_q(G,H)$. Backbone forests in planar graphs are of particular interests. In [22] , we prove that if $G$ is planar and $T$ is a tree of diameter at most 4, then $BB{C}_2(G,T)\le 6$ hence giving an evidence to a conjecture of Broersma et al.  [74] stating that the same holds if $T$ has an arbitrary diameter.

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 [17] , we prove that the Weighted Colouring Problem admits a version of 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  [77] , 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 [58] 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.

On-line colouring

Since many applications, and in particular channel assignment problems, must be solved on-line, we studied on-line colouring algorithms. The most basic and most widespread of them is the greedy algorithm. The largest number of colours that can be given by the greedy algorithm on some graph. is called its Grundy number and is denoted $\Gamma \left(G\right)$. Trivially $\Gamma \left(G\right)\le \Delta \left(G\right)+1$, where $\Delta \left(G\right)$ is the maximum degree of the graph. In [26] , we show that deciding if $\Gamma \left(G\right)\le \Delta \left(G\right)$ is NP-complete. We then show that deciding if $\Gamma \left(G\right)\ge \left|V\right(G\left)\right|-k$ is fixed-parameter tractable with respect to the parameter $k$. We also gave similar complexity results on $b$-colourings, which is a manner of improving colourings on-line.

In [27] , we study a game version of greedy colouring. Given a graph $G$, two players, Alice and Bob, alternate their turns in choosing uncoloured vertices to be coloured. Whenever an uncoloured vertex is chosen, it is coloured by the least positive integer not used by any of its coloured neighbors. Alice's goal is to minimize the total number of colours used in the game, and Bob's goal is to maximize it. The game Grundy number of $G$ is the number of colours used in the game when both players use optimal strategies. It is proved in this paper that the maximum game Grundy number of forests is 3, and the game Grundy number of any partial 2-tree is at most 7.

Enumerating edge-colourings and total colourings

With the success of moderately exponential algorithms, there is an increasing interest for enumeration problems, because of their own interest but also because they might be crucial to solve optimization problems. In [21] , we are interested in computing the number of edge colourings and total colourings of a connected graph. We prove that the maximum number of $k$-edge-colourings of a connected $k$-regular graph on $n$ vertices is $k·{((k-1)!)}^{n/2}$. Our proof is constructive and leads to a branching algorithm enumerating all the $k$-edge-colourings of a connected $k$-regular graph in time $O^{*}\left({((k-1)!)}^{n/2}\right)$ and polynomial space. In particular, we obtain a algorithm to enumerate all the 3-edge-colourings of a connected cubic graph in time $O^{*}\left(2^{n/2}\right)=O^{*}(1.{4143}^n)$ and polynomial space. This improves the running time of $O^{*}(1.{5423}^n)$ of the algorithm of Golovach et al.  [76] . We also show that the number of 4-total-colourings of a connected cubic graph is at most $3·2^{3n/2}$. Again, our proof yields a branching algorithm to enumerate all the 4-total-colourings of a connected cubic graph.

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 a fixed graph $F$ 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 a previous paper, we gave a number of examples of polynomial instances, several NP-completeness proofs as well as a number of conjectures and open problems. In [71] , we conjecture 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 $F$.

Oriented trees in digraphs

Let $f\left(k\right)$ be the smallest integer such that every $f\left(k\right)$-chromatic digraph contains every oriented tree of order $k$. Burr proved $f\left(k\right)\le {(k-1)}^2$ in general, and he conjectured $f\left(k\right)=2k-2$. Burr also proved that every $(8k-7)$-chromatic digraph contains every antidirected tree. We improve both of Burr's bounds. We show [15] that $f\left(k\right)\le k^2/2-k/2+1$ and that every antidirected tree of order $k$ is contained in every $(5k-9)$-chromatic digraph. We also make a conjecture that explains why antidirected trees are easier to handle. It states that if $\left|E\right(D\left)\right|>(k-2)\left|V\right(D\left)\right|$, then the digraph $D$ contains every antidirected tree of order $k$. This is a common strengthening of both Burr's conjecture for antidirected trees and the celebrated Erdős-Sós Conjecture. The analogue of our conjecture for general trees is false, no matter what function $f\left(k\right)$ is used in place of $k-2$. We prove our conjecture for antidirected trees of diameter 3 and present some other evidence for it. Along the way, we show that every acyclic $k$-chromatic digraph contains every oriented tree of order $k$ and suggest a number of approaches for making further progress on Burr's conjecture.