Section: New Results

Graph Theory

Participants : Julio Araújo, Jean-Claude Bermond, Nathann Cohen, Frédéric Giroire, Frédéric Havet, František Kardoš, Ana Karolinna Maia, Dorian Mazauric, Remigiusz Modrzejewski, Leonardo Sampaio, Michel Syska.

Mascotte principally investigates applications in telecommunications via Graph Theory (see other objectives). However it also studies a number of theoretical problems of general interest. Our research mainly focused on graph coloring and some other problems arising from networks problems.

Graph Coloring

Coloring and edge-coloring are two central concepts in Graph Theory. There are many important and long-standing conjectures in these areas. We are trying to make advances towards such conjectures, in particular Steinberg's conjecture, the List coloring Conjecture and the Acyclic Edge-Coloring Conjecture.

We are also interested in coloring problems arising from some practical problems: improper coloring, $L(p,q)$-labeling, directed star arboricity and good edge-labelling. The first two are both motivated by channel assignment and the last two by problems arising in WDM networks. For many practical problems are posed in a dynamic setting, we study on-line coloring and list coloring.

We also study some other variants of coloring like non-repetitive coloring or frugal coloring.

For all the coloring problems, we also consider the associated algorithmic problem, which consists in designing algorithms for finding the minimum number of colors of a coloring of a given graph. Algorithmic results on graph coloring are presented in Section  6.4 .

The most classical notion of coloring (of edges or vertices) is the one of proper coloring, in which we insist on two adjacent elements to have distinct colors. However, it is usual to consider additional constraints,as well as relaxed constraints. For each variant of coloring, one can consider, its list version in which every element $x$ is given a list $L\left(x\right)$ of prescribed colors. A graph is said to be $L$-colorable if is has an $L$-coloring (fulfilling the constraints) such that $x\in L\left(x\right)$ for all element $x$. The choosability of a graph $G$ is the smallest integer $k$ for which $G$ has an $L$-coloring whenever $\left|L\right(x\left)\right|\ge k$ for all elements $x$.

Coloring Graphs with Few Crossings

The famous Four Color Theorem states that every planar graph can be properly colored with 4 colors and Thomassen Five Color Theorem states that the choosability of every planar graph is at most 5. Hence, a natural question is to ask about the chromatic number and choosability of graphs with few crossings. In [38] , we disprove a conjecture of Oporowski and Zhao stating that every graph with crossing number at most 5 and clique number at most 5 is 5-colorable. However, we show that every graph with crossing number at most 4 and clique number at most 5 is 5-colorable. We also show some colorability results on graphs that can be made planar by removing few edges. In particular, we show that if there exists three edges whose removal leaves the graph planar then it is 5-colorable. In [90] , we show that every graph with two crossings is 5-choosable. We also prove that every graph which can be made planar by removing one edge is 5-choosable.

Another famous theorem on planar graphs is the one of Grötzsch, which says that every planar graph with no cycle of length 3 can be properly 3-colored. Steinberg's Conjecture (1976) asserts that a graph with no cycles of length 4 or 5 is 3-colorable. Many approaches have been used towards this conjecture. We considered the following one in which, we relax the constraints on the color classes. Instead of insisting on them be independent sets, we allow them to induce a graph with some bounded degree. A graph $G=(V,E)$ is said to be $(i,j,k)$-colorable if its vertex set can be partitioned into three sets $V_1,V_2,V_3$ such that the graphs $G\left[V_1\right],G\left[V_2\right],G\left[V_3\right]$ induced by the sets $V_1,V_2,V_3$ have maximum degree at most $i,j,k$ respectively. Under this terminology, Steinberg's Conjecture says that every graph with no cycle of length 4 or 5 is $(0,0,0)$-colorable. In [91] , we prove that every graph of $ℱ$ is $(2,1,0)$-colorable and $(4,0,0)$-colorable.

Acyclic, Linear and Frugal Colorings

A classical constraint added to a proper coloring is that at least three colors appears on each cycle, in which case we speak about acyclic coloring. In other words, the graph induced by the elements of any two color classes is a forest. The acyclic chromatic index of a graph $G$, denoted ${\chi }_a^{\text{'}}\left(G\right)$ is the minimum $k$ such that $G$ admits an acyclic edge-coloring with $k$ colors. The famous Acyclic Edge-Coloring Conjecture asserts that ${\chi }_a^{\text{'}}\left(G\right)=\Delta \left(G\right)+2$, where $\Delta \left(G\right)$ is the maximum degree of the graph. In [21] , we conjecture that if $G$ is planar and $\Delta \left(G\right)$ is large enough then ${\chi }_a^{\text{'}}\left(G\right)=\Delta \left(G\right)$. We settle this conjecture for planar graphs with girth at least 5. We also show that ${\chi }_a^{\text{'}}\left(G\right)\le \Delta \left(G\right)+12$ for all planar $G$.

Even stronger constraints are the following: a proper coloring of a graph is 2-frugal (resp. linear) if the graph induced by the elements of any two color classes is of maximum degree 2 (resp. is a forest of paths). In [29] , we improve some bounds on the 2-frugal choosability and linear choosability of graphs with small maximum average degree.

Coloring of Plane Graphs with Constraints on the Faces

We studied several variants of vertex and edge colorings of plane graphs insisting one some constraints on the faces.

A face of a vertex colored plane graph is called loose if the number of colors used on its vertices is at least three. The looseness of a plane graph $G$ is the minimum $k$ such that any surjective $k$-coloring involves a loose face. In [35] , we prove that the looseness of a connected plane graph $G$ equals the maximum number of vertex disjoint cycles in a dual graph $G^{*}$ increased by 2. We also show upper and lower bounds on the looseness of graphs based on the number of vertices, the edge connectivity, and the girth of the dual graph. These bounds improve the result of Negami for the looseness of plane triangulations. We also present infinite classes of graphs where the equalities are attained.

A vertex coloring of a 2-connected plane graph $G$ is a strong parity vertex coloring if for every face $f$ and each color $c$, the number of vertices incident with $f$ colored by $c$ is either zero or odd. Czap et al. [Discrete Math. 311 (2011) 512–520] proved that every 2-connected plane graph has a proper strong parity vertex coloring with at most 118 colors. In [34] , we improve this upper bound for some classes of plane graphs.

A facial parity edge coloring of a connected bridgeless plane graph is such an edge coloring in which no two face-adjacent edges (consecutive edges of a facial walk of some face) receive the same color, in addition, for each face $\alpha$ and each color $c$, either no edge or an odd number of edges incident with $\alpha$ is colored with $c$. From Vizing's theorem it follows that every 3-connected plane graph has a such coloring with at most ${\Delta }^{*}+1$ colors, where ${\Delta }^{*}$ is the size of the largest face. In [33] we prove that any connected bridgeless plane graph has a facial parity edge coloring with at most 92 colors.

A sequence $r_1,r_2,\cdots ,r_{2n}$ such that $r_i=r_{n+i}$ for all $1\le i\le n$, is called a repetition. A sequence $S$ is called non-repetitive if no block (i.e. subsequence of consecutive terms of $S$) is a repetition. Let $G$ be a graph whose edges are colored. A trail is called non-repetitive if the sequence of colors of its edges is non-repetitive. If $G$ is a plane graph, a facial non-repetitive edge-coloring of $G$ is an edge-coloring such that any facial trail (i.e. trail of consecutive edges on the boundary walk of a face) is non-repetitive. We denote ${\pi }_f^{\text{'}}\left(G\right)$ the minimum number of colors of a facial non-repetitive edge-coloring of $G$. In [41] , we show that ${\pi }_f^{\text{'}}\left(G\right)\le 8$ for any plane graph $G$. We also get better upper bounds for ${\pi }_f^{\text{'}}\left(G\right)$ in the cases when $G$ is a tree, a plane triangulation, a simple 3-connected plane graph, a hamiltonian plane graph, an outerplanar graph or a Halin graph. The bound 4 for trees is tight.

Improper Coloring

In [85] and [48] , we study a coloring problem motivated by a practical frequency assignment problem and up to our best knowledge new. In wireless networks, a node interferes with the other nodes the level of interference depending on numerous parameters: distance between the nodes, geographical topography, obstacles, etc. We model this with a weighted graph $G$ where the weights on the edges represent the noise (interference) between the two end-nodes. The total interference in a node is then the sum of all the noises of the nodes emitting on the same frequency. A weighted $t$-improper $k$-coloring of $G$ is a $k$-coloring of the nodes of $G$ (assignment of $k$ frequencies) such that the interference at each node does not exceed some threshold $t$. The Weighted Improper Coloring problem, that we consider here consists in determining the weighted $t$-improper chromatic number defined as the minimum integer $k$ such that $G$ admits a weighted $t$-improper $k$-coloring. We also consider the dual problem, denoted the Threshold Improper Coloring problem, where given a number $k$ of colors (frequencies) we want to determine the minimum real $t$ such that $G$ admits a weighted $t$-improper $k$-coloring. We show that both problems are NP-hard and first present general upper bounds; in particular we show a generalization of Lovász's Theorem for the weighted $t$-improper chromatic number. We then show how to transform an instance of the Threshold Improper Coloring problem into another equivalent one where the weights are either 1 or $M$, for a sufficient big value $M$. Motivated by the original application, we study a special interference model on various grids (square, triangular, hexagonal) where a node produces a noise of intensity 1 for its neighbors and a noise of intensity 1/2 for the nodes that are at distance 2. Consequently, the problem consists of determining the weighted $t$-improper chromatic number when $G$ is the square of a grid and the weights of the edges are 1, if their end nodes are adjacent in the grid, and 1/2 otherwise. Finally, we model the problem using linear integer programming, propose and test heuristic and exact Branch-and-Bound algorithms on random cell-like graphs, namely the Poisson-Voronoi tessellations.

On-line Coloring

Several on-line algorithms producing colorings have been designed. The most basic and most widespread one 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. Determning the Grundy number of a graph is NP-hard even for $P_5$-free graphs, while it is poynomial-time solvable for $P_4$-free graphs. In [19] , we define a new class of graphs, namely the fat-extended $P_4$-laden graphs, which intersects the class of $P_5$-free graphs and strictly contains the one of $P_4$-free. We show a polynomial-time algorithm to determine the Grundy number of such graphs. It implies that the Grundy number can be computed in polynomial time for most graph classes defined in terms of containing few $P_4$'s: $P_4$-reducible, extended $P_4$-reducible, $P_4$-sparse, extended $P_4$-sparse, ...

In [94] , we study a game version of greedy coloring. Given a graph $G=(V,E)$, two players, Alice and Bob, alternate their turns in choosing uncolored vertices to be colored. Whenever an uncolored vertex is chosen, it is colored by the least positive integer not used by any of its colored neighbors. Alice's goal is to minimize the total number of colors used in the game, and Bob's goal is to maximize it. The game Grundy number of $G$ is the number of colors 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. We also gave some complexity results on $b$-colorings, which is a manner of improving colorings on-line [43] .

Other Results on Graph Coloring

In [18] , we aim at characterizing the class of graphs that admit a good edge-labelling. Such graphs are interesting, as they correspond to set of requests in UPP-digraphs (those in which there is at most one dipath from a vertex to another) for which the minimum number of wavelengths is equal to the maximum load. This implies that the problem can be solved efficiently. First, we exhibit infinite families of graphs for which no good edge-labelling can be found. We then show that deciding if a graph admits a good edge-labelling is NP-complete. Finally, we give large classes of graphs admitting a good edge-labelling: C3 -free outerplanar graphs, planar graphs of girth at least 6, subcubic C3, K2, 3 -free graphs.

A wheel is a graph formed by a chordless cycle and a vertex that has at least three neighbors in the cycle. We prove in [83] that every 3-connected graph that does not contain a wheel as a subgraph is in fact minimally 3-connected. We prove that every graph that does not contain a wheel as a subgraph is 3-colorable. We were then told that this result was already proved by Thomassen, though with a different proof.

Gallai-Hasse-Roy-Vitaver Theorem states that every $n$-chromatic digraph contains a directed path of order $n$. 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 that $f\left(k\right)\le {(k-1)}^2$ and conjectured $f\left(k\right)=2n-2$. In [84] , we give some sufficient conditions for an $n$-chromatic digraphs to contains some oriented tree. In particular, we show that every acyclic $n$-chromatic digraph contains every oriented tree of order $n$. We also show that $f\left(k\right)\le k^2/2-k/2+1$. Finally, we consider the existence of antidirected trees in digraphs. We prove that every antidirected tree of order $k$ is contained in every $(5k-9)$-chromatic digraph. We conjecture 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 generalizes Burr's conjecture for antidirected trees and the celebrated Erdős-Sós Conjecture. We give some evidences for our conjecture to be true.

Matchings and Independent Sets

Matchings and independent sets are important substructures which appears in many problems. In particular, color classes of vertex-colorings and edge-colorings are independent sets and matchings, respectively.

In [45] , we show that every (sub)cubic $n$-vertex graph with sufficiently large girth has fractional chromatic number at most $2.2978$ which implies that it contains an independent set of size at least $0.4352n$. Our bound on the independence number is valid to random cubic graphs as well as it improves existing lower bounds on the maximum cut in cubic graphs with large girth.

In [39] , we show that every cubic bridgeless graph $G$ has at least $2^{\left|V\right(G\left)\right|/3656}$ perfect matchings. This confirms an old and celebrated conjecture of Lovász and Plummer in the 1970's. This improves the first superlinear bound given in [40] .

Hypergraphs

Hypergraphs, also called set systems, are a natural generalization of graphs. In a graph an edge is set of two vertices, while in a hypergraph an edge is a set of any size. It turns out to be an important notion in database theory. A digraph is $\alpha$-acyclic if it can be reduced to the null hypergraph by succesively removing either a vertex which in at most one edge or an edge included in another. It is one of the possible generalizations of forest to hypergraphs. The $\alpha$-arboricity of a hypergraph $H$ is the minimum number of $\alpha$-acyclic hypergraphs that partition the edge set of $H$. In [23] , the $\alpha$-arboricity of the complete 3-uniform (every edge is a set of 3 vertices) hypergraph is determined completely.

In [80] , we generalize the concept of line digraphs to line dihypergraphs. We give some general properties in particular concerning connectivity parameters of dihypergraphs and their line dihypergraphs, like the fact that the arc connectivity of a line dihypergraph is greater than or equal to that of the original dihypergraph. Then we show that the De Bruijn and Kautz dihypergraphs (which are among the best known bus networks) are iterated line digraphs. Finally we give short proofs that they are highly connected.

Miscellaneous

Zagreb Indices

The first and second Zagreb indices of a graph are defined by $M_1={\sum }_{v\in V\left(G\right)}d{\left(v\right)}^2$ and $M_2={\sum }_{uv\in E\left(G\right)}d\left(u\right)d\left(v\right)$, respectively. They are used in chemistry where it represents properties of molecules. In [17] , we present some classes of graphs with prescribed degrees that satisfy $M_1/n\le M_2/m$, where $M_1$ and $M_2$ are the first and second Zagreb indices. We also prove that for any $\Delta \ge 5$, there is an infinite family of graphs of maximum degree $\Delta$ such that the inequality is false. Moreover, we give alternative and slightly shorter proof of this inequality for trees and unicyclic graphs.

Induced Decomposition

An induced $H$-decomposition of a graph $G$ is a partition $(E_1,\cdots ,E_k)$ of its edge set $E\left(G\right)$, such that the graph induced by each $E_i$, $1\le i\le k$, is a copy of $H$. Bondy and Szwarcfiter asked for the maximum number $ex(n,H)$ of edges in a graph on $n$ vertices which admits an induced $H$-decomposition. In [13] , we prove that for every non-empty graph $H$, $ex(n,H)=n(n-1)/2-o\left(n^2\right)$.