Section: New Results

Graph theory

Participants : Nathann Cohen, Guillaume Ducoffe, Frédéric Havet, William Lochet, Nicolas Nisse.

Coati also studies theoretical problems in graph theory. If some of them are directly motivated by applications (see Subsection 7.3.3), others are more fundamental. In particular, we are putting an effort on understanding better directed graphs (also called digraphs) and partionning problems, and in particular colouring problems. We also try to better the understand the many relations between orientation and colourings. We study various substructures and partitions in (di)graphs. For each of them, we aim at giving sufficient conditions that guarantee its existence and at determining the complexity of finding it.

Substructures in digraphs

Arc-disjoint branching flows

The concept of arc-disjoint flows in networks was introduced by Bang-Jensen and Bessy [Theoret. Comput. Science 526, 2014]. This 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 [15], we prove that for every fixed integer $k\ge 2$ 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-time 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 $n-k\left(n\right)$.

Subdivision of oriented cycles

An oriented cycle is an orientation of a undirected cycle. In [43], [27], we first show that for any oriented cycle $C$, there are digraphs containing no subdivision of $C$ (as a subdigraph) and arbitrarily large chromatic number. In contrast, we show that for any cycle $C$ with two blocks, every strongly connected digraph with sufficiently large chromatic number contains a subdivision of $C$. This settles a conjecture of Addario-Berry et al. [J. Combin. Theory B, 97, 2007]. More generally, we conjecture that this result holds for any oriented cycle. As a further evidence, we prove this conjecture for the antidirected cycle on four vertices (in which two vertices have out-degree 2 and two vertices have in-degree 2).

Colourings and partitioning (di)graphs

2-partitions of digraphs

A $k$-partition of a (di)graph $D$ is a partition of $V\left(D\right)$ into $k$ disjoint sets. Let ${\mathbb{P}}_1,{\mathbb{P}}_2$ be two (di)graph properties, then a $({\mathbb{P}}_1,{\mathbb{P}}_2)$-partition of a (di)graph $D$ is a 2-partition $(V_1,V_2)$ where $V_1$ induces a (di)graph with property ${\mathbb{P}}_1$ and $V_2$ a (di)graph with property ${\mathbb{P}}_2$. In [14], [13] and [38], [37], we give a complete characterization for the complexity of $({\mathbb{P}}_1,mathbb{P}_2)$-partition problems when ${\mathbb{P}}_1,{\mathbb{P}}_2$ are one of the following standard properties: acyclic, complete, independent (no arcs), oriented (no directed 2-cycle), semicomplete, tournament, symmetric (if two vertices are adjacent, then they induce a directed 2-cycle), strongly connected, connected, minimum out-degree at least 1, minimum in-degree at least 1, minimum semi-degree at least 1, minimum degree at least 1, having an out-branching, having an in-branching. We also investigate the influence of strong connectivity of the input digraph on this complexity. In particular, we show that some NP-complete probems become polynomial-time solvable when restricted to strongly connected input digraphs.

$\chi$-bounded families of oriented graphs

A famous conjecture of Gyárfás and Sumner states for any tree $T$ and integer $k$, if the chromatic number of a graph is large enough, either the graph contains a clique of size $k$ or it contains $T$ as an induced subgraph. In [57], we discuss some results and open problems about extensions of this conjecture to oriented graphs. We conjecture that for every oriented star $S$ and integer $k$, if the chromatic number of a digraph is large enough, either the digraph contains a clique of size $k$ or it contains $S$ as an induced subgraph. As an evidence, we prove that for any oriented star $S$, every oriented graph with sufficiently large chromatic number contains either a transitive tournament of order 3 or $S$ as an induced subdigraph. We then study for which sets $𝒫$ of orientations of $P_4$ (the path on four vertices) similar statements hold. We establish some positive and negative results.

Locally irregular decompositions of subcubic graphs

A graph $G$ is locally irregular if every two adjacent vertices of $G$ have different degrees. A locally irregular decomposition of $G$ is a partition $E_1,...,E_k$ of $E\left(G\right)$ such that each $G\left[E_i\right]$ is locally irregular. Not all graphs admit locally irregular decompositions, but for those who are decomposable, in that sense, it was conjectured by Baudon, Bensmail, Przybylo and Wozniak that they decompose into at most 3 locally irregular graphs. Towards that conjecture, it was recently proved by Bensmail, Merker and Thomassen that every decomposable graph decomposes into at most 328 locally irregular graphs. In [39], we focus on locally irregular decompositions of subcubic graphs, which form an important family of graphs in this context, as all non-decomposable graphs are subcubic. As a main result, we prove that decomposable subcubic graphs decompose into at most 5 locally irregular graphs, and only 4 when the maximum average degree is less than 12/5. We then consider weaker decompositions, where subgraphs can also include regular connected components, and prove the relaxations of the conjecture above for subcubic graphs.

Orientation and edge-weigthing inducing colouring

An orientation of a graph $G$ is proper if two adjacent vertices have different indegrees. The proper-orientation number of a graph $G$ is the minimum maximum indegree of a proper orientation of $G$. In a previous paper, we raise the question whether the proper orientation number of a planar graph is bounded. In [12], we prove that every cactus admits a proper orientation with maximum indegree at most 7. We also prove that the bound 7 is tight by showing a cactus having no proper orientation with maximum indegree less than 7. We also prove that any planar claw-free graph has a proper orientation with maximum indegree at most 6 and that this bound can also be attained.

Sum-distinguishing edge-weightings

A $k$-edge-weighting of a graph $G$ is an application from $E\left(G\right)$ into $\{1,\cdots ,k\}$. An edge-weigthing is sum-distinguishing if for every two adajcent vertices $u$ and $v$, the sum of weights of edges incident to $u$ is distinct from the sum of of weights of edges incident to $v$. The celebrated 1-2-3-Conjecture (raised in 2004 by Karoński, Luczak and Thomason) asserts that every connected graph (except $K_2$, the complete graph on two vertices) admits a sum-distinguishing 3-edge-weighting. This conjecture attracted much attention and many variants are now studied. We study several of them.

In [58], we study the existence of sum-distinguishing injective $\left|E\right(G\left)\right|$-edge-weightings. We conjecture that such an edge-weighting always exists (except from $K_2)$. We prove this conjecture for some classes of graphs, such as trees and regular graphs. In addition, for some other classes of graphs, such as 2-degenerate graphs and graphs with maximum average degree at most 3, we prove that, provided we use a constant number of additional edge weights, the desired edge-weighting always exists. Our investigations are strongly related to several aspects of the well-known 1-2-3 Conjecture and the Antimagic Labelling Conjecture.

One of the variants consists in considering total-labelling rather than edge-weighting. A $k$-total-weighting of a graph $G$ is an application from $V\left(G\right)\cup E\left(G\right)$ into $\{1,\cdots ,k\}$. An edge-weigthing is sum-distinguishing if for every two adajcent vertices $u$ and $v$, the sum of of weights of $u$ and the edges incident to $u$ is distinct from the sum of weights of $v$ and the edges incident to $v$. The 1-2 Conjecture raised by Przybylo, lo and Wozniak in 2010 asserts that every undirected graph admits a 2-total-weighting (both vertices and eedges receives weights) such that the sums of weights "incident" to the vertices yield a proper vertex-colouring. Following several recent works bringing related problems and notions (such as the well-known 1-2-3 Conjecture, and the notion of locally irregular decompositions) to digraphs, we introduce in [40] and study several variants of the 1-2 Conjecture for digraphs. For every such variant, we raise conjectures concerning the number of weights necessary to obtain a desired total-weighting in any digraph. We verify some of these conjectures, while we obtain close results towards the ones that are still open.

Colouring game

We wish to motivate the problem of finding decentralized lower-bounds on the complexity of computing a Nash equilibrium in graph games. While the centralized computation of an equilibrium in polynomial time is generally perceived as a positive result, this does not reflect well the reality of some applications where the game serves to implement distributed resource allocation algorithms, or to model the social choices of users with limited memory and computing power. As a case study, we investigate in [31] on the parallel complexity of a game-theoretic variation of graph colouring. These “colouring games" were shown to capture key properties of the more general welfare games and Hedonic games. On the positive side, it can be computed a Nash equilibrium in polynomial-time for any such game with a local search algorithm. However, the algorithm is time-consuming and it requires polynomial space. The latter questions the use of colouring games in the modeling of information-propagation in social networks. We prove that the problem of computing a Nash equi- librium in a given colouring game is PTIME-hard, and so, it is unlikely that one can be computed with an efficient distributed algorithm. The latter brings more insights on the complexity of these games.

Identifying codes

Let $G$ be a graph $G$. The neighborhood of a vertex $v$ in $G$, denoted by $N\left(v\right)$, is the set of vertices adjacent to $v$ i $G$. It closed neighborhood is the set $N\left[v\right]=N\left(v\right)\cup \left\{v\right\}$. A set $C\subseteq V\left(G\right)$ is an identifying code in $G$ if (i) for all $v\in V\left(G\right)$, $N\left[v\right]\cap C\ne \varnothing$, and (ii) for all $u,v\in V\left(G\right)$, $N\left[u\right]\cap C\ne N\left[v\right]\cap C$. The problem of finding low-density identifying codes was introduced in [Karpovsky et al., IEEE Trans. Inform. Theory 44, 1998] in relation to fault diagnosis in arrays of processors. Here the vertices of an identifying code correspond to controlling processors able to check themselves and their neighbors. Thus the identifying property guarantees location of a faulty processor from the set of “complaining” controllers. Identifying codes are also used in [Ray et al., IEEE Journal on Selected Areas in Communications 22, 2004] to model a location detection problem with sensor networks.

Particular interest was dedicated to grids as many processor networks have a grid topology. There are three types of regular infinite grids in the plane, namely the hexagonal grids, the square grids and the triangular grids. In [26], [42], we study the square grid ${𝒮}_k)$ with infinite width and bounded height $k$. We prove that the minimum density of an identifying code in ${𝒮}_k$ is at least $\frac{7}{20}+\frac{1}{20k}$ and at most $\frac{7}{20}+\frac{3}{10k}$. We also establish that the minimum density of a code in an infinite square grid of height 3 is $\frac{7}{18}$. In [49], [30], we study the minimum density $d^{*}\left({𝒯}_k\right)$ of the triangular grid ${𝒮}_k)$ with infinite width and bounded height $k$. We prove that $d^{*}\left(T_k\right)=\frac{1}{4}+\frac{1}{4k}$ for every odd $k$ and $\frac{1}{4}+\frac{1}{4k}\le d^{*}\left(T_k\right)\le \frac{1}{4}+\frac{1}{2k}$ for every even $k$. We also prove $d^{*}\left(T_2\right)=\frac{1}{2}$ and $d^{*}\left(T_4\right)=d^{*}\left(T_6\right)=\frac{1}{3}$. All these proofs are made using the discharging method, which seems not have been very rarely used for such problems whereas it applies very well.