## Section: New Results

### Graph theory

Participants : Julien Bensmail, Frédéric Havet, William Lochet, Nicolas Nisse, Fionn Mc Inerney, Stéphane Pérennes, Bruce Reed.

Coati studies theoretical problems in graph theory. If some of them are directly motivated by applications, others are more fundamental.

#### Interval number in cycle convexity

Recently, Araujo et al. [Manuscript in preparation, 2017] introduced the notion of Cycle Convexity of graphs.
In their seminal work, they studied the graph convexity parameter called hull number for this new graph convexity they proposed, and they presented some of its applications in Knot theory.
Roughly, the *tunnel number* of a knot embedded in a plane is upper bounded by the hull number of a corresponding planar 4-regular graph in cycle convexity.
In [4], we go further in the study of this new graph convexity and we study the interval number of a graph in cycle convexity.
This parameter is, alongside the hull number, one of the most studied parameters in the literature about graph convexities.
Precisely, given a graph $G$, its *interval number* in cycle convexity, denoted by $CCIHN\left(G\right)$, is the minimum cardinality of a set $S\subseteq V\left(G\right)$ such that every vertex $w\in V\left(G\right)\setminus S$ has two distinct neighbors $u,v\in S$ such that $u$ and $v$ lie in same connected component of $G\left[S\right]$, i.e. the subgraph of $G$ induced by the vertices in $S$.

In [4] we provide bounds on $CCIHN\left(G\right)$ and its relations to other graph convexity parameters, and explore its behaviour on grids. Then, we present some hardness results by showing that deciding whether $CCIHN\left(G\right)\le k$ is NP-complete, even if $G$ is a split graph or a bounded-degree planar graph, and that the problem is W[2]-hard in bipartite graphs when $k$ is the parameter. As a consequence, we obtain that $CCIHN\left(G\right)$ cannot be approximated up to a constant factor in the classes of split graphs and bipartite graphs (unless $P=NP$).

On the positive side, we present polynomial-time algorithms to compute $CCIHN\left(G\right)$ for outerplanar graphs, cobipartite graphs and interval graphs. We also present fixed-parameter tractable (FPT) algorithms to compute it for $(q,q-4)$-graphs when $q$ is the parameter and for general graphs $G$ when parameterized either by the treewidth or the neighborhood diversity of $G$.

Some of our hardness results and positive results are not known to hold for related graph convexities and domination problems. We hope that the design of our new reductions and polynomial-time algorithms can be helpful in order to advance in the study of related graph problems.

#### Steinberg-like theorems for backbone colouring

A function $f:V\left(G\right)\to \{1,...,k\}$ is a (proper) $k$-colouring of $G$ if $\left|f\right(u)-f(v\left)\right|\ge 1$, for every edge $uv\in E\left(G\right)$. The *chromatic number* $\chi \left(G\right)$ is the smallest integer $k$ for which there exists a proper $k$-colouring of $G$.
Given a graph $G$ and a subgraph $H$ of $G$, a circular $q$-backbone $k$-colouring $f$ of $(G,H)$ is a $k$-colouring of $G$ such that $q\le \left|c\right(u)-c(v\left)\right|\le k-q$, for each edge $uv\in E\left(H\right)$. The *circular $q$-backbone chromatic number* of a graph pair $(G,H)$, denoted ${\mathrm{CBC}}_{q}(G,H)$, is the minimum $k$ such that $(G,H)$ admits a circular $q$-backbone $k$-colouring.
Steinberg conjectured that if $G$ is planar and $G$ contains no cycles on 4 or 5 vertices, then $\chi \left(G\right)\le 3$. If this conjecture is correct, then one could deduce that ${\mathrm{CBC}}_{2}(G,H)\le 6$, for any $H\subseteq G$. In [5], we first show that if $G$ is a planar graph containing no cycle on 4 or 5 vertices and $H\subseteq G$ is a forest, then ${\mathrm{CBC}}_{2}(G,H)\le 7$. Then, we prove that if $H\subseteq G$ is a forest whose connected components are paths, then ${\mathrm{CBC}}_{2}(G,H)\le 6$.

#### Homomorphisms of planar signed graphs and absolute cliques

Homomorphisms are an important topic in graph theory, as example the chromatic number of a graph $G$ is the minimum $k$ such that $G$ maps onto the complete graph ${K}_{k}$. A signed graph $(G,\Sigma )$ is a (simple) graph with sign function $\Sigma E\left(G\right)\to \{-1,1\}$. A closed-walk is unbalanced if it has an odd number of negative edges, it is balanced otherwise. Homomorphisms of signed graphs are mapping that preserve adjacency and balance of cycles. Naserasr, Rollova and Sopena (Journal of Graph Theory 2015) posed the important question of finding out the minimum size $k$ such that any planar signed graph $(G,\Sigma )$ admits a homomorphism to a signed graph with $k$ vertices. The question can be seen as the counterpart of the 4 color theorem which implies that any palnar graph maps onto ${K}_{4}$. It is known that if this minimum value is equal to 10, then every planar signed graph maps to a particular unique signed graph $({P}^{+}9,{\Gamma}^{+})$ with 10 vertices. A graph $G$ is an underlying absolute signed clique if there exists a signed graph $(G,\Sigma )$ which does not admit any homomorphism to any signed graph $(H,\Pi )$ with $\left|V\right(H\left)\right|<\left|V\right(G\left)\right|$. In [66] we characterize all underlying absolute signed planar cliques up to spanning subgraph inclusion. Furthermore, we show that every signed planar graph having underlying graphs obtained by (repeated, finite) $k$-clique sums ($k\le 3$) of underlying absolute signed planar cliques admits a homomorphism to $({P}^{+}9,{\Gamma}^{+})$. Based on this evidence, we conjecture that every planar signed graph admits a homomorphism to $(P+9,{\Gamma}^{+})$.

#### Edge-partitioning a graph into paths: the Barát-Thomassen conjecture

In 2006, Barát and Thomassen conjectured that there is a function $f$ such that, for every fixed tree $T$ with $t$ edges, every $f\left(t\right)$-edge-connected graph with its number of edges divisible by $t$ has a partition of its edges into copies of $T$. We recently proved this conjecture with Merker [69].

The path case of the Barát-Thomassen conjecture (i.e $\forall k,m=\left|E\right|\phantom{\rule{0.166667em}{0ex}}mod\phantom{\rule{0.166667em}{0ex}}k=0$ there exists $f\left(k\right)$ such that if the connectivity of $G$ is larger than $f\left(k\right)$ then $G$ can be partitionned into ${P}_{k}$) has also been studied, notably by Thomassen [80], [81], [82], and had been solved by Botler, Mota, Oshiro and Wakabayashi [70]. In [15] we propose an alternative proof of the path case with a weaker hypothesis: Namely, we prove that there is a function $f$ such that every 24-edge-connected graph with minimum degree $f\left(k\right)$ has an edge-partition into paths of length $k$. We also show that 24 can be dropped to 4 when the graph is Eulerian.

#### Some Aspects of Arbitrarily Partitionable Graphs

An $n$-graph $G$ is arbitrarily partitionable (AP) if, for every partition of $n$ as $n={n}_{1}+...+{n}_{p}$, there is a partition $({V}_{1},...,{V}_{p})$ of $V\left(G\right)$ such that for $i=1,...,p$ $G\left[{V}_{i}\right]$ is connected and $|{V}_{i}|={n}_{i}$. The property of being AP is related to other well-known graph notions, such as perfect matchings and Hamiltonian cycles (obviously Hamiltonian graph is AP), with which it shares several properties. In [65] This work we studying two aspects of AP graphs.

On the one hand, we consider the algorithmic aspects.
We first establish the *NP*-hardness of the problem of partitioning a graph into connected subgraphs following a given sequence, for various new graph classes of interest.
We then prove that the problem of deciding whether a graph is AP is *NP*-hard for several classes of graphs, confirming a conjecture of Barth and Fournier.

On the other hand, we consider the weakening of APness to sufficient conditions for Hamiltonicity. While previous works have suggested that such conditions can sometimes indeed be weakened, we point out cases for which this is not true. This is done by considering conditions for Hamiltonicity involving squares of graphs, and claw- and net-free graphs.

#### Incident Sum problems and the 1-2-3 Conjecture

How can one distinguish the adjacent vertices of a graph through an edge-weighting? In the last decades, this question has been attracting increasing attention,
which resulted in the active field of distinguishing labelings. One of its most popular problems is the one where neighbours must be distinguishable via their incident sums of weights.
An edge-weighting verifying this is said to be *proper*. The popularity of this notion arises mainly due to the influence of the famous 1-2-3 Conjecture (posed by Karoński, Łuczak and Thomason), which claims that proper weightings with weights in $\{1,2,3\}$ exist for graphs with no isolated edge.

The questions that we study aim at solving or at progressing toward the solution of the 1-2-3 conjecture and similar problems.

In [8] we study locally irregular decompositions of sub-cubic graphs. A graph $G$ is locally irregular if every two adjacent vertices of $G$ have different degrees (this corresponds to a uniform weight). A locally irregular decomposition of $G$ is a partition ${E}_{1},\cdots ,{E}_{k}$ of the edge set $E\left(G\right)$ such that each $G\left[{E}_{i}\right]$ is locally irregular. Not all graphs admit locally irregular decompositions, but for those who are decomposable, it was conjectured by Baudon, Bensmail, Przybyło and Woźniak that the decomposition uses 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. Our work focuses on the case of sub-cubic graphs, which form an important family of graphs in this context, as all non-decomposable graphs are sub-cubic. As a main result, we prove that decomposable sub-cubic graphs decompose into at most 5 locally irregular graphs, and at most 4 when the maximum average degree is less than $\frac{12}{5}$. We then consider weaker decomposition, where subgraphs can also include regular connected components, and prove the relaxations of the conjecture above for sub-cubic graphs.

In [9] we pursue recent works generalizing "Neighbour Sum problems" (e.g. the well-known 1-2-3 Conjecture, or the notion of locally irregular decomposition) to digraphs. We introduce and study several variants of the 1-2 Conjecture for digraphs and for every such variant, we state 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 solution of the ones that are still open.

In [10] we study a variant of the classical 1-2-3 Conjecture.
This conjecture asks whether every graph but ${K}_{2}$ can be 3-edge-weighted
so that every two adjacent vertices $u$ and $v$ can be distinguished via the sum of their incident weights, that is the incident sums of $u$ and $v$ differ by at least 1.
In this work we investigate the consequences on the 1-2-3 Conjecture of requiring a
stronger distinction condition, that is requiring the incident sums to differ by at least 2.
Our conjecture is that every graph but ${K}_{2}$ admits a 5-edge-weighting permitting to distinguish the adjacent vertices in this stronger way. We prove 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 when the graph is a tree.

In [11] we prove a 1-2-3-4 result for the 1-2-3 Conjecture in 5-regular graphs. Currently the best-known result toward te 1-2-3 conjecture 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 of the problem (.i.e in our context total means that both the vertices and the edges are assigned weights). Our work, 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 adjacent vertices.

In [63]
we investigate another aspect of edge weighting that allow to distinguish adjacent vertices (we shall call them *proper*). Namely we study the minimum number of distinct neighbourhood sums we can produce using such proper weightings. Clearly, this minimum number is bounded below by the chromatic number $\chi \left(G\right)$ of $G$.
When using weights in $Z$, we show that we can always produce proper edge-weightings generating $\chi \left(G\right)$ distinct sums but in the peculiar case where $G$ is a balanced bipartite graph, in which case exactly $\chi \left(G\right)+1$ distinct sums have to be generated.
When using $k$ consecutive weights $1,...,k$, we provide both lower and upper bounds, as a function of the maximum degree $\Delta $,
on the maximum least number of sums that can be generated for a graph with maximum degree $\Delta $.
For trees, which, in general, admit neighbour-sum-distinguishing 2-edge-weightings,
we prove that this maximum, when using weights 1 and 2, is of order $2{log}_{2}\Delta $.
Finally, we also establish the *NP*-hardness of several decision problems related to these questions.

The 1-2-3 Conjecture has recently been investigated from a decompositional angle, via so-called locally irregular decompositions, which are edge-partitions into locally irregular subgraphs. Through several recent studies, it was shown that this concept is quite related to the 1-2-3 Conjecture. However, the full connection between all those concepts was not clear. In [55], we propose an approach that generalizes all concepts above, involving coloured weights and sums. As a consequence, we get another interpretation of several existing results related to the 1-2-3 Conjecture. We also propose new related conjectures, to which we give some support.

#### Identifying codes

For $G$ a graph or a digraph, let $\mathrm{id}\left(G\right)$ be the minimum size of an identifying code of $G$ if one exists, and $\mathrm{id}\left(G\right)=+\infty $ otherwise. For a graph $G$, let $\mathrm{idor}\left(G\right)$ be the minimum of $\mathrm{id}\left(D\right)$ overall orientations $D$ of $G$. In [20], we give some lower and upper bounds on $\mathrm{idor}\left(G\right)$. In particular, we show that $\mathrm{idor}\left(G\right)\le 3/2\mathrm{id}\left(G\right)$ for every graph $G$. We also show that computing $\mathrm{idor}\left(G\right)$ is NP-hard, while deciding whether $\mathrm{idor}\left(G\right)\le \left|V\right(G\left)\right|-k$ is polynomial-time solvable for every fixed integer $k$.