## Section: New Results

### Graph theory

Participants : Julien Bensmail, Guillaume Ducoffe, Frédéric Havet, William Lochet, Nicolas Nisse, Bruce Reed.

Coati 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 (di)graphs

We study various conditions that ensure a (di)graph to contain certain substructures.

In [17], we study the question of finding a set of $k$ vertex-disjoint cycles (resp. directed cycles) of distinct lengths in a given graph (resp. digraph). In the context of undirected graphs, we prove that, for every $k\ge 1$, every graph with minimum degree at least $\frac{{k}^{2}+5k-2}{2}$ has $k$ vertex-disjoint cycles of different lengths, where the degree bound is the best possible. We also consider other cases such as when the graph is triangle-free, or the $k$ cycles are required to have different lengths modulo some value $r$. In the context of directed graphs, we consider a conjecture of Lichiardopol concerning the least minimum out-degree required for a digraph to have $k$ vertex-disjoint directed cycles of different lengths. We verify this conjecture for tournaments, and, by using the probabilistic method, for some regular digraphs and digraphs of small order.

A *$({k}_{1}+{k}_{2})$-bispindle* is the union of ${k}_{1}$ $(x,y)$-dipaths and ${k}_{2}$ $(y,x)$-dipaths, all these dipaths being pairwise internally disjoint. Recently, Cohen et al. showed that for every $(1,1)$-bispindle $B$, there exists an integer $k$ such that every strongly connected digraph with chromatic number greater than $k$ contains a subdivision of $B$. In [24], we investigate generalisations of this result by first showing constructions of strongly connected digraphs with large chromatic number without any $(3,0)$-bispindle or $(2,2)$-bispindle. Then we show that strongly connected digraphs with large chromatic number contains a $(2,1)$-bispindle, where at least one of the $(x,y)$-dipaths and the $(y,x)$-dipath are long.

Let $\mathscr{H}$ be a family of graphs and let $d$ be large enough. For every $d$-regular graph $G$, we study the existence of a spanning $\mathscr{H}$-free subgraph of $G$ with large minimum degree. This problem is well understood if $\mathscr{H}$ does not contain bipartite graphs. In [35] we provide asymptotically tight results for many families of bipartite graphs such as cycles or complete bipartite graphs. To prove these results, we study a locally injective analogue of the question.

An *even pair* (resp. *odd pair*) in a graph is a pair of non-adjacent vertices such that every chordless path between them has even (resp. odd) length.
Even and odd pairs are important tools in the study of perfect graphs and were instrumental in the proof of the Strong Perfect Graph Theorem. We suggest that such pairs impose a lot of structure also in arbitrary, not just perfect graphs. To this end, we show in [36] that the presence of even or odd pairs in graphs imply a special structure of the stable set polytope. In fact, we give a polyhedral characterization of even and odd pairs.

#### Colourings and partitioning (di)graphs

##### Colouring graphs with constraints on connectivity

A graph $G$ has maximal local edge-connectivity $k$ if the maximum number of edge-disjoint paths between every pair of distinct vertices $x$ and $y$ is at most $k$. We prove in [11] Brooks-type theorems for $k$-connected graphs with maximal local edge-connectivity $k$, and for any graph with maximal local edge-connectivity 3. We also consider several related graph classes defined by constraints on connectivity. In particular, we show that there is a polynomial-time algorithm that, given a 3-connected graph $G$ with maximal local connectivity 3, outputs an optimal colouring for $G$. On the other hand, we prove, for $k\ge 3$, that $k$-colourability is NP-complete when restricted to minimally $k$-connected graphs, and 3-colourability is NP-complete when restricted to $\left(k-1\right)$-connected graphs with maximal local connectivity $k$. Finally, we consider a parameterization of $k$-colourability based on the number of vertices of degree at least $k+1$, and prove that, even when $k$ is part of the input, the corresponding parameterized problem is FPT.

##### 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.

Towards the 1-2-3-Conjecture, the best-known result to date is due to Kalkowski, Karoński and Pfender, who proved that it holds when relaxed to 5-edge-weightings. Their proof builds upon a weighting algorithm designed by Kalkowski for a total version (where also the vertices are weighted) of the problem. In [67], we present new mechanisms for using Kalkowski's algorithm in the context of the 1-2-3 Conjecture. As a main result we prove that every 5-regular graph admits a 4-edge-weighting that permits to distinguish its adjacent vertices via their incident sums.

In [66], we investigate the consequences on the 1-2-3 Conjecture of requiring a stronger distinction condition. Namely, we consider two adjacent vertices distinguished when their incident sums differ by at least 2. As a guiding line, we conjecture that every graph with no connected component isomorphic to ${K}_{2}$ admits a 5-edge-weighting permitting to distinguish the adjacent vertices in this stronger way.
We verify this conjecture for several classes of graphs, including bipartite graphs and cubic graphs. We then consider algorithmic aspects, and show that it is *NP*-complete to determine the smallest $k$ such that a given bipartite graph admits such
a $k$-edge-weighting. In contrast, we show that the same problem can be solved in polynomial time for a given tree.

In [13], we consider the following question, which stands as a directed analogue of the 1-2-3 Conjecture: Given any digraph $D$ with no arc $\overrightarrow{uv}$ verifying ${d}^{+}\left(u\right)={d}^{-}\left(v\right)=1$, is it possible to weight the arcs of $D$ with weights among $\{1,2,3\}$ so that, for every arc $\overrightarrow{uv}$ of $D$, the sum of incident weights out-going from $u$ is different from the sum of incident weights in-coming to $v$? We answer positively to this question, and investigate digraphs for which even the weights among $\{1,2\}$ are sufficient. In relation with the so-called 1-2 Conjecture, we also consider a total version of the problem, which we prove to be false. Our investigations turn to have interesting relations with open questions related to the 1-2-3 Conjecture.

In [21], we study the following question: Is it always possible to injectively assign the weights $1,...,\left|E\right(G\left)\right|$ to the edges of any given graph $G$ (with no component isomorphic to ${K}_{2}$) so that every two adjacent vertices of $G$ get distinguished by their sums of incident weights? One may see this question as a combination of the well-known 1-2-3 Conjecture and the Antimagic Labelling Conjecture. We exhibit evidence that this question might be true. Benefiting from the investigations on the Antimagic Labelling Conjecture, we first point out that several classes of graphs, such as regular graphs, indeed admit such assignments. We then show that trees also do, answering a recent conjecture of Arumugam, Premalatha, Bača and Semaničová-Feňovčíková. Towards a general answer to the question above, we then prove that claimed assignments can be constructed for any graph, provided we are allowed to use some number of additional edge weights. For some classes of sparse graphs, namely 2-degenerate graphs and graphs with maximum average degree 3, we show that only a small (constant) number of such additional weights suffices.

##### Variants of vertex- or edge-colouring

A colouring of a graph $G$ is *properly connected* if every two vertices of $G$ are the ends of a properly coloured path. In [57], [47], we study the *complexity* of computing the *proper connection number* (minimum number of colours in a properly connected colouring) for edge and vertex colourings, in undirected and directed graphs, respectively.
First we disprove some conjectures of Magnant et al. (2016) on characterizing the strong digraphs with *proper arc connection number* at most two. Then, we prove that deciding whether a given digraph has proper arc connection number at most two is NP-complete. Furthermore, we show that there are infinitely many such digraphs with no even-length dicycle.
We initiate the study of proper vertex connectivity in digraphs and we prove similar results as for the arc version. Finally, we present polynomial-time recognition algorithms for *bounded-treewidth* graphs and *bipartite* graphs with *proper edge connection number* at most two.

A graph is *locally irregular* if no two adjacent vertices have the same degree. The *irregular chromatic index* ${\chi}_{\mathrm{irr}}^{\text{'}}\left(G\right)$ of a graph $G$ is the smallest number of locally irregular subgraphs needed to edge-decompose $G$. Not all graphs have such a decomposition, but Baudon, Bensmail, Przybyło, and Woźniak conjectured that if $G$ can be decomposed into locally irregular subgraphs, then ${\chi}_{\mathrm{irr}}^{\text{'}}\left(G\right)\le 3$. In support of this conjecture, Przybyło showed that ${\chi}_{\mathrm{irr}}^{\text{'}}\left(G\right)\le 3$ holds whenever $G$ has minimum degree at least ${10}^{10}$.
In [19] we prove that every bipartite graph $G$ which is not an odd length path satisfies ${\chi}_{\mathrm{irr}}^{\text{'}}\left(G\right)\le 10$. This is the first general constant upper bound on the irregular chromatic index of bipartite graphs. Combining this result with Przybyło's result, we show that ${\chi}_{\mathrm{irr}}^{\text{'}}\left(G\right)\le 328$ for every graph $G$ which admits a decomposition into locally irregular subgraphs. Finally, we show that ${\chi}_{\mathrm{irr}}^{\text{'}}\left(G\right)\le 2$ for every 16-edge-connected bipartite graph $G$.

An $(m,n)$-coloured mixed graph is a mixed graph with arcs assigned one of $m$ different colours and edges one of $n$ different colours. A homomorphism of an $(m,n)$-coloured mixed graph $G$ to an $(m,n)$-coloured mixed graph $H$ is a vertex mapping such that if $uv$ is an arc (edge) of colour $c$ in $G$, then $f\left(u\right)f\left(v\right)$ is also an arc (edge) of colour $c$. The $(m,n)$-coloured mixed chromatic number, denoted ${\chi}_{m,n}\left(G\right)$, of an $(m,n)$-coloured mixed graph $G$ is the order of a smallest homomorphic image of $G$. An $(m,n)$-clique is an $(m,n)$-coloured mixed graph $C$ with ${\chi}_{m,n}\left(C\right)=\left|V\left(C\right)\right|$. In [16], we study the structure of $(m,n)$-cliques. We show that almost all $(m,n)$-coloured mixed graphs are $(m,n)$-cliques, prove bounds for the order of a largest outerplanar and planar $(m,n)$-clique and resolve an open question concerning the computational complexity of a decision problem related to $(0,2)$-cliques. Additionally, we explore the relationship between ${\chi}_{1,0}$ and ${\chi}_{0,2}$.

An edge colouring of a graph $G$ is called acyclic if it is proper and every cycle contains at least three colours. We show in [33] that for every $\epsilon >0$, there exists a $g=g\left(\epsilon \right)$ such that if $G$ has maximum degree $\Delta $ and girth at least $g$ then $G$ admits an acyclic edge colouring with $(1+\epsilon )\Delta +O\left(1\right)$ colours.

#### 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 $v$ in $G$.
Its *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.

A particular interest was dedicated to grids as many processor networks have a grid topology. There are several types of standard regular infinite grids, in particular the hexagonal grids, the square grids, the triangular grids and the king grids. For such graphs $G$, the problem consists in finding the minimum density ${d}^{*}\left(G\right)$ of an identifying code of $G$.

In [26], we study the infinite triangular grid ${T}_{k}$ with $k$ rows. We show ${d}^{*}\left({T}_{1}\right)={d}^{*}\left({T}_{2}\right)=1/2$, ${d}^{*}\left({T}_{3}\right)={d}^{*}\left({T}_{4}\right)=1/3$, ${d}^{*}\left({T}_{5}\right)=3/10$, ${d}^{*}\left({T}_{6}\right)=1/3$ and ${d}^{*}\left({T}_{k}\right)=1/4+1/\left(4k\right)$ for all odd $k\ge 7$. In addition, we show that $1/4+1/\left(4k\right)\le {d}^{*}\left({T}_{k}\right)\le 1/4+1/\left(2k\right)$ for all even $k\ge 8$.

In [27], we study the density of king grids which are strong product of two paths. We show that for every king grid $G$, ${d}^{*}\left(G\right)\ge 2/9$. In addition, we show this bound is attained only for king grids which are strong products of two infinite paths. Given $k\ge 3$, we denote by ${K}_{k}$ the (infinite) king strip with $k$ rows. We prove that ${d}^{*}\left({K}_{3}\right)=1/3$, ${d}^{*}\left({K}_{4}\right)=5/16$, ${d}^{*}\left({K}_{5}\right)=4/15$ and ${d}^{*}\left({K}_{6}\right)=5/18$. We also prove that $2/9+8/81k\le {d}^{*}\left({K}_{k}\right)\le 2/9+4/9k$ for every $k\ge 7$.

#### Miscellaneous

##### A proof of the Barát-Thomassen conjecture

The Barát-Thomassen conjecture asserts that for every tree $T$ on $m$ edges, there exists a constant ${k}_{T}$ such that every ${k}_{T}$-edge-connected graph with size divisible by $m$ can be edge-decomposed into copies of $T$. So far this conjecture has only been verified when $T$ is a path or when $T$ has diameter at most 4. In [18], we prove the full statement of the conjecture.

##### Recursively partitionable graphs

A connected graph $G$ is said to be *arbitrarily partitionable* (AP for short) if for every partition $({n}_{1},...,{n}_{p})$ of $\left|V\right(G\left)\right|$ there exists a partition $({V}_{1},...,{V}_{p})$ of $V\left(G\right)$ such that each ${V}_{i}$ induces a connected subgraph of $G$ on ${n}_{i}$ vertices. Some stronger versions of this property were introduced, namely the ones of being *online arbitrarily partitionable* and *recursively arbitrarily partitionable* (OL-AP and R-AP for short, respectively), in which the subgraphs induced by a partition of $G$ must not only be connected but also fulfil additional conditions. In [14], we point out some structural properties of OL-AP and R-AP graphs with connectivity 2. In particular, we show that deleting a cut pair of these graphs results in a graph with a bounded number of components, some of whom have a small number of vertices. We obtain these results by studying a simple class of 2-connected graphs called *balloons*

##### On oriented cliques with respect to push operation

An oriented graph is a directed graph without any directed cycle of length at most 2. An oriented clique is an oriented graph whose non-adjacent vertices are connected by a directed 2-path. To push a vertex $v$ of a directed graph $\overrightarrow{G}$ is to change the orientations of all the arcs incident to $v$. A push clique is an oriented clique that remains an oriented clique even if one pushes any set of vertices of it. We show in [20] that it is NP-complete to decide if an undirected graph is the underlying graph of a push clique or not. We also prove that a planar push clique can have at most 8 vertices and provide an exhaustive list of planar push cliques.

##### On $q$-power cycles in cubic graphs

In the context of a conjecture of Erdős and Gyárfás, we consider in [15], for any $q\ge 2$, the existence of $q$-power cycles (*i.e.* with length a power of $q$) in cubic graphs. We exhibit constructions showing that, for every $q\ge 3$, there exist arbitrarily large cubic graphs with no $q$-power cycles. Concerning the remaining case $q=2$ (which corresponds to the conjecture of Erdős and Gyárfás), we show that there exist arbitrarily large cubic graphs whose only 2-power cycles have length 4 only, or 8 only.

##### How to determine if a random graph with a fixed degree sequence has a giant component

For a fixed degree sequence $\mathcal{D}=({d}_{1},...,{d}_{n})$, let $G\left(\mathcal{D}\right)$ be a uniformly chosen (simple) graph on $\{1,\cdots ,n\}$ where the vertex $i$ has degree ${d}_{i}$. In [34] we determine whether $G\left(\mathcal{D}\right)$ has a giant component with high probability, essentially imposing no conditions on $\mathcal{D}$. We simply insist that the sum of the degrees in $\mathcal{D}$ which are not 2 is at least $\lambda \left(n\right)$ for some function $\lambda $ going to infinity with $n$. This is a relatively minor technical condition, and when $\mathcal{D}$ does not satisfy it, both the probability that $G\left(\mathcal{D}\right)$ has a giant component and the probability that $G\left(\mathcal{D}\right)$ has no giant component are bounded away from 1.

##### A proof of the Erdős-Sands-Sauer-Woodrow conjecture

A very nice result of Barany and Lehel asserts that every finite subset $X$ of ${R}^{d}$ can be covered by $f\left(d\right)$ $X$-boxes (i.e. each box has two antipodal points in $X$). As shown by Gyárfás and Pálvőlgyi this result would follow from the following conjecture : If a tournament admits a partition of its arc set into $k$ partial orders, then its domination number is bounded in terms of $k$. This question is in turn implied by the Erdős-Sands-Sauer-Woodrow conjecture : If the arcs of a tournament $T$ are colored with $k$ colors, there is a set $X$ of at most $g\left(k\right)$ vertices such that for every vertex $v$ of $T$, there is a monochromatic path from $X$ to $v$. We give in [69] a short proof of this statement. We moreover show that the general Sands-Sauer-Woodrow conjecture (which as a special case implies the stable marriage theorem) is valid for directed graphs with bounded stability number. This conjecture remains however open.