Section: New Results

Graph and digraph theory

Participants : Julien Bensmail, Frédéric Havet, Nicolas Nisse, Stéphane Pérennes, Francois Dross, Fionn Mc Inerney, Thi Viet Ha Nguyen, Nathann Cohen.

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

We are putting an effort on understanding better directed graphs (also called digraphs) and partitioning problems, and in particular colouring problems. We also try to better the understand the many relations between orientations 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.

To ease the reading, we split our results in this section into several subsections dedicated to particular topics.

Graph and digraph colourings

Distinguishing labellings and the 1-2-3 Conjecture

We are interested in several distinguishing labelling (or edge-weighting) problems, where the general aim, given a graph, is to label the edges in such a way that certain properties are fulfilled. The main problem we have been considering is the 1-2-3 Conjecture, which claims that every connected graph different from $K_2$ admits a labelling with $1,2,3$ such that no two adjacent vertices are incident to the same sum of weights. Some of our latest results provide evidence towards the 1-2-3 Conjecture. We also investigated questions inspired from the conjecture, such that the role of the weights $1,2,3$ in the statement of the conjecture, the deep connection with proper vertex-colourings and other standard notions of graph theory.

A 1-2-3-4 result for the 1-2-3 Conjecture in 5-regular graphs

To date, the best-known result towards the 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. In [23], 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.

On $\{a,b\}$-edge-weightings of bipartite graphs with odd $a,b$

For any $S\subset \mathbb{Z}$ we say that a graph $G$ has the $S$-property if there exists an $S$-edge-weighting $w:E\left(G\right)\to S$ such that for any pair of adjacent vertices $u,v$ we have ${\sum }_{e\in E\left(v\right)}w\left(e\right)\ne {\sum }_{e\in E\left(u\right)}w\left(e\right)$, where $E\left(v\right)$ and $E\left(u\right)$ are the sets of edges incident to $v$ and $u$, respectively. In general, deciding if a graph $G$ has the $\{a,b\}$-property is NP-complete for every $a,b$. This question is open for bipartite graphs however. The only known results of this sort are that bipartite graphs without the $\{1,2\}$-property can be recognized easily, and similarly for 2-connected bipartite graphs without the $\{0,1\}$-property. In [28], we focus on $\{a,a+2\}$-edge-weightings where $a\in \mathbb{Z}$ is odd. We show that a 2-connected bipartite graph has the $\{a,a+2\}$-property if and only if it is not a so-called odd multi-cactus. In the case of trees, we show that only one case is pathological. That is, we show that all trees have the $\{a,a+2\}$-property for odd $a\ne -1$, while there is an easy characterization of trees without the $\{-1,1\}$-property.

1-2-3 Conjecture in Digraphs: More Results and Directions

When arc-weighting a digraph, there are, at each vertex, two sums of incident weights: the in-coming sum ${\sigma }^{-}$ and the out-going sum ${\sigma }^{+}$. Thus, there are many ways for generalizing the 1-2-3 Conjecture to digraphs. In the recent years, four main variants have been considered, where, for every arc $\overrightarrow{uv}$, it is required that one of ${\sigma }^{-}\left(u\right),{\sigma }^{+}\left(u\right)$ is different from one of ${\sigma }^{-}\left(v\right),{\sigma }^{+}\left(v\right)$. All of these four variants are well understood, except for the one where, for every arc $\overrightarrow{uv}$, it is required that ${\sigma }^{-}\left(u\right)\ne {\sigma }^{+}\left(v\right)$. Regarding this version, Horňak, Przybyło and Woźniak recently proved that almost every digraph can be 4-arc-weighted so that, for every arc $\overrightarrow{uv}$, the sum of weights incoming to $u$ is different from the sum of weights outgoing from $v$. They conjectured a stronger result, namely that the same statement with 3 instead of 4 should also be true. We verify this conjecture in [73]. This work takes place in a recent “quest” towards a directed version of the 1-2-3 Conjecture, the variant above being one of the last introduced ones. We take the occasion of this work to establish a summary of all results known in this field, covering known upper bounds, complexity aspects, and choosability. On the way we prove additional results which were missing in the whole picture. We also mention the aspects that remain open.

Edge Weights and Vertex Colours: Minimizing Sum Count

Put differently, the 1-2-3 Conjecture asks whether, via weights with very low magnitude, we can “encode” a proper vertex-colouring of any graph. Note, however, that we do not care about whether such a result colouring is optimal, i.e., whether its number of colours is close to the chromatic number. In [22], we investigate the minimum number of distinct sums/colours we can produce via a neighbour-sum-distinguishing edge-weighting of a given graph $G$, and the role of the assigned weights in that context. Clearly, this minimum number is bounded below by the chromatic number $\chi \left(G\right)$ of $G$. When using weights of $\mathbb{Z}$, we show that, in general, we can produce neighbour-sum-distinguishing edge-weightings generating $\chi \left(G\right)$ distinct sums, except in the peculiar case where $G$ is a balanced bipartite graph, in which case $\chi \left(G\right)+1$ distinct sums can be generated. These results are best possible. 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.

On Minimizing the Maximum Color for the 1-2-3 Conjecture

In the line of the previous investigation, one way to get some sort of progress is to design proper labellings where the maximum color of a vertex is as small as possible. In [64], we investigate the consequences of labeling graphs as in the 1-2-3 Conjecture when it is further required to make the maximum resulting color as small as possible. We first investigate the hardness of determining the minimum maximum color by a labeling for a given graph, which we show is NP-complete in the class of bipartite graphs but polynomial-time solvable in the class of graphs with bounded treewidth. We then provide bounds on the minimum maximum color that can be generated both in the general context, and for particular classes of graphs. Finally, we study how using larger labels permits to reduce the maximum color.

Decomposing degenerate graphs into locally irregular subgraphs

A (undirected) graph is locally irregular if no two of its adjacent vertices have the same degree. A decomposition of a graph $G$ into $k$ locally irregular subgraphs is a partition $E_1,...,E_k$ of $E\left(G\right)$ into $k$ parts each of which induces a locally irregular subgraph. Not all graphs decompose into locally irregular subgraphs; however, it was conjectured that, whenever a graph does, it should admit such a decomposition into at most three locally irregular subgraphs. This conjecture was verified for a few graph classes in recent years. It was introduced because it was noticed that, in some contexts, there are connections between locally irregular decompositions and the 1-2-3 Conjecture. In [63], we consider the decomposability of degenerate graphs with low degeneracy. Our main result is that decomposable $k$-degenerate graphs decompose into at most $3k+1$ locally irregular subgraphs, which improves on previous results whenever $k\le 9$. We improve this result further for some specific classes of degenerate graphs, such as bipartite cacti, $k$-trees, and planar graphs. Although our results provide only little progress towards the leading conjecture above, the main contribution of this work is rather the decomposition schemes and methods we introduce to prove these results.

A general decomposition theory for the 1-2-3 Conjecture and locally irregular decompositions

In [21], we propose an approach encapsulating locally irregular decompositions and proper labelings. As a consequence, we get another interpretation of several existing results related to the 1-2-3 Conjecture. We also come up with new related conjectures, to which we give some support.

Decomposability of graphs into subgraphs fulfilling the 1-2-3 Conjecture

In particular, one of the side problems we run into is decomposing graphs into subgraphs verifying the 1-2-3 Conjecture. In [29], we prove that every $d$-regular graph, $d\ge 2$, can be decomposed into at most 2 subgraphs (without isolated edges) fulfilling the 1-2-3 Conjecture if $d\notin \{10,11,12,13,15,17\}$, and into at most 3 such subgraphs in the remaining cases. Additionally, we prove that in general every graph without isolated edges can be decomposed into at most 24 subgraphs fulfilling the 1–2–3 Conjecture, improving the previously best upper bound of 40. Both results are partly based on applications of the Lovász Local Lemma.

On the 2-edge-coloured chromatic number of grids

The oriented (2-edge-coloured, respectively) chromatic number ${\chi }_{\mathrm{o}}\left(G\right)$ (${\chi }_2\left(G\right)$, respectively) of an undirected graph $G$ is defined as the maximum oriented (2-edge-coloured, respectively) chromatic number of an orientation (signature, respectively) of $G$. Although the difference between ${\chi }_{\mathrm{o}}\left(G\right)$ and ${\chi }_2\left(G\right)$ can be arbitrarily large, there are, however, contexts in which these two parameters are quite comparable. In [24], we compare the behaviour of these two parameters in the context of (square) grids. While a series of works has been dedicated to the oriented chromatic number of grids, we are not aware of any work dedicated to their 2-edge-coloured chromatic number. We investigate this throughout this paper. We show that the maximum 2-edge-coloured chromatic number of a grid lies between 8 and 11. We also focus on 2-row grids and 3-row grids, and exhibit bounds on their 2-edge-coloured chromatic number, some of which are tight. Although our results indicate that the oriented chromatic number and the 2-edge-coloured chromatic number of grids are close in general, they also show that these parameters may differ, even for easy instances.

From light edges to strong edge-colouring of 1-planar graphs

A strong edge-colouring of an undirected graph $G$ is an edge-colouring where every two edges at distance at most 2 receive distinct colours. The strong chromatic index of $G$ is the least number of colours in a strong edge-colouring of $G$. A conjecture of Erdős and Nešetřil, stated back in the 80's, asserts that every graph with maximum degree $\Delta$ should have strong chromatic index at most roughly $1.25{\Delta }^2$. Several works in the last decades have confirmed this conjecture for various graph classes. In particular, lots of attention have been dedicated to planar graphs, for which the strong chromatic index decreases to roughly $4\Delta$, and even to smaller values under additional structural requirements. In [26], we initiate the study of the strong chromatic index of 1-planar graphs, which are those graphs that can be drawn on the plane in such a way that every edge is crossed at most once. We provide constructions of 1-planar graphs with maximum degree $\Delta$ and strong chromatic index roughly $6\Delta$. As an upper bound, we prove that the strong chromatic index of a 1-planar graph with maximum degree $\Delta$ is at most roughly $24\Delta$ (thus linear in $\Delta$). In the course of proving the latter result, we prove, towards a conjecture of Hudák and Šugerek, that 1-planar graphs with minimum degree 3 have edges both of whose ends have degree at most 29.

Pushable chromatic number of graphs with degree constraints

Pushable homomorphisms and the pushable chromatic number ${\chi }_p$ of oriented graphs were introduced by Klostermeyer and MacGillivray in 2004. They notably observed that, for any oriented graph $\overrightarrow{G}$, we have ${\chi }_p\left(\overrightarrow{G}\right)\le {\chi }_o\left(\overrightarrow{G}\right)\le 2{\chi }_p\left(\overrightarrow{G}\right)$, where ${\chi }_o\left(\overrightarrow{G}\right)$ denotes the oriented chromatic number of $\overrightarrow{G}$. This stands as first general bounds on ${\chi }_p$. This parameter was further studied in later works.

In [71], we consider the pushable chromatic number of oriented graphs fulfilling particular degree conditions. For all $\Delta \ge 29$, we first prove that the maximum value of the pushable chromatic number of an oriented graph with maximum degree $\Delta$ lies between $2^{\frac{\Delta }{2}-1}$ and $(\Delta -3)·(\Delta -1)·2^{\Delta -1}+2$ which implies an improved bound on the oriented chromatic number of the same family of graphs. For subcubic oriented graphs, that is, when $\Delta \le 3$, we then prove that the maximum value of the pushable chromatic number is 6 or 7. We also prove that the maximum value of the pushable chromatic number of oriented graphs with maximum average degree less than 3 lies between 5 and 6. The former upper bound of 7 also holds as an upper bound on the pushable chromatic number of planar oriented graphs with girth at least 6.

Graph and digraph decompositions

Edge-partitioning a graph into paths: beyond 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 $T$. This conjecture was recently verified in  [81] by, in particular, some members of Coati . In [27], we further focus on the path case of the Barát-Thomassen conjecture. Before the aforementioned general proof was announced, several successive steps towards the path case of the conjecture were made, notably by Thomassen  [94], [95], [96], until this particular case was totally solved by Botler, Mota, Oshiro and Wakabayashi  [84]. Our goal in this work was to 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 $f\left(t\right)$ has an edge-partition into paths of length $t$ whenever $t$ divides the number of edges. We also show that 24 can be dropped to 4 when the graph is eulerian.

Constrained ear decompositions in graphs and digraphs.

Ear decompositions of graphs are a standard concept related to several major problems in graph theory like the Traveling Salesman Problem. For example, the Hamiltonian Cycle Problem, which is notoriously NP-complete, is equivalent to deciding whether a given graph admits an ear decomposition in which all ears except one are trivial (i.e. of length 1). On the other hand, a famous result of Lovász states that deciding whether a graph admits an ear decomposition with all ears of odd length can be done in polynomial time. In [38], we study the complexity of deciding whether a graph admits an ear decomposition with prescribed ear lengths. We prove that deciding whether a graph admits an ear decomposition with all ears of length at most $\ell$ is polynomial-time solvable for all fixed positive integer $\ell$. On the other hand, deciding whether a graph admits an ear decomposition without ears of length in $ℱ$ is NP-complete for any finite set $ℱ$ of positive integers. We also prove that, for any $k\ge 2$, deciding whether a graph admits an ear decomposition with all ears of length $0modk$ is NP-complete.

We also consider the directed analogue to ear decomposition, which we call handle decomposition, and prove analogous results : deciding whether a digraph admits a handle decomposition with all handles of length at most $\ell$ is polynomial-time solvable for all positive integer $\ell$; deciding whether a digraph admits a handle decomposition without handles of length in $ℱ$ is NP-complete for any finite set $ℱ$ of positive integers (and minimizing the number of handles of length in $ℱ$ is not approximable up to $n(1-ϵ)$); for any $k\ge 2$, deciding whether a digraph admits a handle decomposition with all handles of length $0modk$ is NP-complete. Also, in contrast with the result of Lovász, we prove that deciding whether a digraph admits a handle decomposition with all handles of odd length is NP-complete. Finally, we conjecture that, for every set $𝒜$ of integers, deciding whether a digraph has a handle decomposition with all handles of length in $𝒜$ is NP-complete, unless there exists $h\in \mathbb{N}$ such that $𝒜=\{1,\cdots ,h\}$.

Substructures in graphs and digraphs

Subdivisions in Digraphs of Large Out-Degree or Large Dichromatic Number

In 1985, Mader conjectured the existence of a function $f$ such that every digraph with minimum out-degree at least $f\left(k\right)$ contains a subdivision of the transitive tournament of order $k$. This conjecture is still completely open, as the existence of $f\left(5\right)$ remains unknown. In this paper, we show that if $D$ is an oriented path, or an in-arborescence (i.e., a tree with all edges oriented towards the root) or the union of two directed paths from $x$ to $y$ and a directed path from $y$ to $x$, then every digraph with minimum out-degree large enough contains a subdivision of $D$. Additionally, we study Mader's conjecture considering another graph parameter. The dichromatic number of a digraph $D$ is the smallest integer $k$ such that $D$ can be partitioned into $k$ acyclic subdigraphs. We show in [16] that any digraph with dichromatic number greater than $4m(n-1)$ contains every digraph with $n$ vertices and $m$ arcs as a subdivision.

Bipartite spanning sub(di)graphs induced by 2-partitions

For a given 2-partition $(V_1,V_2)$ of the vertices of a (di)graph $G$, we study properties of the spanning bipartite subdigraph $BG(V_1,V_2)$ of $G$ induced by those arcs/edges that have one end in each $V_i,i\in \{1,2\}$. In [20], we determine, for all pairs of non-negative integers $k_1,k_2$, the complexity of deciding whether $G$ has a 2-partition $(V_1,V_2)$ such that each vertex in $V_i$ (for $i\in \{1,2\}$) has at least $k_i$ (out-)neighbours in $V_{3-i}$. We prove that it is NP-complete to decide whether a digraph $D$ has a 2-partition $(V_1,V_2)$ such that each vertex in $V_1$ has an out-neighbour in $V_2$ and each vertex in $V_2$ has an in-neighbour in $V_1$. The problem becomes polynomially solvable if we require $D$ to be strongly connected. We give a characterization of the structure of NP-complete instances in terms of their strong component digraph. When we want higher in-degree or out-degree to/from the other set the problem becomes NP-complete even for strong digraphs. A further result is that it is NP-complete to decide whether a given digraph $D$ has a 2-partition $(V_1,V_2)$ such that $BD(V_1,V_2)$ is strongly connected. This holds even if we require the input to be a highly connected eulerian digraph.

Metric Dimension: from Graphs to Oriented Graphs

The metric dimension $MD\left(G\right)$ of an undirected graph $G$ is the cardinality of a smallest set of vertices that allows, through their distances to all vertices, to distinguish any two vertices of $G$. Many aspects of this notion have been investigated since its introduction in the 70's, including its generalization to digraphs.

In [42], [43], we study, for particular graph families, the maximum metric dimension over all strongly-connected orientations, by exhibiting lower and upper bounds on this value. We first exhibit general bounds for graphs with bounded maximum degree. In particular, we prove that, in the case of subcubic $n$-node graphs, all strongly-connected orientations asymptotically have metric dimension at most $\frac{n}{2}$, and that there are such orientations having metric dimension $\frac{2n}{5}$. We then consider strongly-connected orientations of grids. For a torus with $n$ rows and $m$ columns, we show that the maximum value of the metric dimension of a strongly-connected Eulerian orientation is asymptotically $\frac{nm}{2}$ (the equality holding when $n,m$ are even, which is best possible). For a grid with $n$ rows and $m$ columns, we prove that all strongly-connected orientations asymptotically have metric dimension at most $\frac{2nm}{3}$, and that there are such orientations having metric dimension $\frac{nm}{2}$.

Bio-informatics motivated problems

Overlaying a hypergraph with a graph with bounded maximum degree

A major problem in structural biology is the characterization of low resolution structures of macro-molecular assemblies. One subproblem of this very difficult question is to determine the plausible contacts between the subunits (e.g. proteins) of an assembly, given the lists of subunits involved in all the complexes. This problem can be conveniently modelled by graphs and hypergraphs. Let $G$ and $H$ be respectively a graph and a hypergraph defined on a same set of vertices, and let $F$ be a fixed graph. We say that $G$ $F$-overlays a hyperedge $S$ of $H$ if $F$ is a spanning subgraph of the subgraph of $G$ induced by $S$, and that it $F$-overlays $H$ if it $F$-overlays every hyperedge of $H$. Motivated by structural biology, we study in [68] the computational complexity of two problems. The first problem, $(\Delta \le k)$ $F$-Overlay, consists in deciding whether there is a graph with maximum degree at most $k$ that $F$-overlays a given hypergraph $H$. It is a particular case of the second problem Max $(\Delta \le k)$ $F$-Overlay, which takes a hypergraph $H$ and an integer $s$ as input, and consists in deciding whether there is a graph with maximum degree at most $k$ that $F$-overlays at least $s$ hyperedges of $H$. We give a complete polynomial/NP-complete dichotomy for the Max $(\Delta \le k)$ $F$-Overlay problems depending on the pairs $(F,k)$, and establish the complexity of $(\Delta \le k)$ $F$-Overlay for many pairs $(F,k)$.