This thesis is comprised of three parts. In the first one,a channel assignment problem posed by Alcatel is modelledas a graph colouring problem: a graph is k-improperlyl-colourable if, given l colours,every vertex can be assigned a colour such that each colourclass induces a subgraph of maximum degree at most k.Several problems are studied:...
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July 5, 2006 (v1)PublicationUploaded on: March 25, 2023
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March 2004 (v1)Report
Improper choosability of planar graphs has been widely studied. In particular, Skrekovski investigated the smallest integer $g_k$ such that every planar graph of girth at least $g_k$ is $k$-improper $2$-choosable. He proved that $6\leq g_1\leq 9$; $5\leq g_2\leq 7$; $5\leq g_3\leq 6$ and $\forall k\geq 4, g_k=5$. In this paper, we study the...
Uploaded on: December 3, 2022 -
2008 (v1)Report
The process number of a digraph has been introduced as a tool to study rerouting issues in WDM networks. We consider the recognition and the characterization of (di)graphs with process number at most two.
Uploaded on: February 22, 2023 -
July 6, 2011 (v1)Journal article
We introduce the process number of a digraph as a tool to study rerouting issues in \wdm networks. This parameter is closely related to the vertex separation (or pathwidth). We consider the recognition and the characterization of (di)graphs with small process number. In particular, we give a linear time algorithm to recognize (and process)...
Uploaded on: December 4, 2022 -
February 20, 2008 (v1)Journal article
A plane graph is l-facially k-colourable if its vertices can be coloured with k colours such that any two distinct vertices on a facial segment of length at most l are coloured differently. We prove that every plane graph is 3-facially 11-colourable. As a consequence, we derive that every 2-connected plane graph with maximum face-size at most 7...
Uploaded on: December 3, 2022 -
2006 (v1)Report
We are interested in the relation between the pathwidth of a biconnected outerplanar graph and the pathwidth of its (geometric) dual. Bodlaender and Fomin, after having proved that the pathwidth of every biconnected outerplanar graph is always at most twice the pathwidth of its (geometric) dual plus two, conjectured that there exists a constant...
Uploaded on: December 3, 2022 -
2005 (v1)Conference paper
For any graph $G$, the $k$-improper chromatic number $χ ^k(G)$ is the smallest number of colours used in a colouring of $G$ such that each colour class induces a subgraph of maximum degree $k$. We investigate the ratio of the $k$-improper chromatic number to the clique number for unit disk graphs and random unit disk graphs to extend results of...
Uploaded on: March 25, 2023 -
January 2009 (v1)Journal article
A graph is k-improperly -colourable if its vertices can be partitioned into parts such that each part induces a subgraph of maximum degree at most k. A result of Lovasz a states that for any graph G, such a partition exists if l is at least (Δ(G)+1)/(k+1) . When k = 0, this bound can be reduced by Brooks' Theorem, unless G is complete or an odd...
Uploaded on: December 4, 2022 -
February 2012 (v1)Journal article
An L(p,1)-labeling of a graph is a function f from the vertex set to the positive integers such that |f(x) − f(y)| ≥ p if dist(x, y) = 1 and |f(x) − f(y)| ≥ 1 if dist(x, y) = 2, where dist(x,y) is the distance between the two vertices x and y in the graph. The span of an L(p,1)- labeling f is the difference between the largest and the smallest...
Uploaded on: December 3, 2022 -
January 20, 2008 (v1)Conference paper
An $L(2,1)$-labelling of a graph is a function $f$ from the vertex set to the positive integers such that $|f(x)-f(y)|\geq 2$ if $dist(x,y)=1$ and $|f(x)-f(y)|\geq 1$ if $dist(x,y)=2$, where $dist(u,v)$ is the distance between the two vertices~$u$ and~$v$ in the graph $G$. The \emph{span} of an $L(2,1)$-labelling $f$ is the difference between...
Uploaded on: December 4, 2022 -
2009 (v1)Journal article
International audience
Uploaded on: December 3, 2022 -
2007 (v1)Journal article
We are interested in the relation between the pathwidth of a biconnected outerplanar graph and the pathwidth of its (geometric) dual. Bodlaender and Fomin [3], after having proved that the pathwidth of every biconnected outerplanar graph is always at most twice the pathwidth of its (geometric) dual plus two, conjectured that there exists a...
Uploaded on: December 4, 2022 -
July 10, 2007 (v1)Conference paper
Bermond-Thomassen conjecture says that a digraph of minimum out-degree at least 2r−1, r >=1, contains at least r vertex-disjoint directed cycles. Thomassen proved that it is true when r=2, but it is still open for larger values of r, even when restricted to (regular) tournaments. In this paper, we present two proofs of this conjecture for...
Uploaded on: February 28, 2023 -
2005 (v1)Report
For any graph $G$, the $k$-improper chromatic number $\chi^k(G)$ is the smallest number of colours used in a colouring of $G$ such that each colour class induces a subgraph of maximum degree $k$. We investigate the ratio of the $k$-improper chromatic number to the clique number for unit disk graphs and random unit disk graphs to generalise...
Uploaded on: December 3, 2022 -
2007 (v1)Report
Motivated by a satellite communications problem, we consider a generalised colouring problem on unit disk graphs. A colouring is k -improper if no vertex receives the same colour as k +1 of its neighbours. The k -improper chromatic number chi_k (G) is the least number of colours needed in a k -improper colouring of a graph G. The main sub ject...
Uploaded on: February 28, 2023 -
October 2013 (v1)Journal article
Numerous sophisticated local algorithm were suggested in the literature for various fundamental problems. Notable examples are the MIS and (∆+1)-coloring algorithms by Barenboim and Elkin [6], by Kuhn [22], and by Panconesi and Srinivasan [34], as well as the O(∆ 2)-coloring algorithm by Linial [28]. Unfortunately, most known local algorithms...
Uploaded on: March 25, 2023 -
2007 (v1)Report
In this paper, we study the notion of circular choosability recently introduced by Mohar and Zhu. First, we provide a negative answer to a question of Zhu about circular cliques. We next prove that, for every graph G, cch(G) = O( ch(G) + ln |V(G)| ). We investigate a generalisation of circular choosability, circular f-choosability, when f is a...
Uploaded on: December 3, 2022 -
2009 (v1)Journal article
We study circular choosability, a notion recently introduced by Mohar and by Zhu. First, we provide a negative answer to a question of Zhu about circular cliques. We next prove that cch(G) = O(ch(G) + ln |V(G)|) for every graph G. We investigate a generalisation of circular choosability, the circular f-choosability, where f is a function of the...
Uploaded on: December 4, 2022 -
April 2010 (v1)Journal article
A vertex coloring of a plane graph is ℓ-facial if every two distinct vertices joined by a facial walk of length at most ℓ receive distinct colors. It has been conjectured that every plane graph has an ℓ-facial coloring with at most 3ℓ+1 colors. We improve the currently best known bound and show that every plane graph has an ℓ-facial coloring...
Uploaded on: December 3, 2022 -
July 16, 2012 (v1)Conference paper
We generalize the classical cow-path problem [7, 14, 38, 39] into a question that is relevant for collective foraging in animal groups. Specifically, we consider a setting in which k identical (probabilistic) agents, initially placed at some central location, collectively search for a treasure in the two-dimensional plane. The treasure is...
Uploaded on: March 25, 2023 -
2007 (v1)Report
In this paper, we study the notion of circular choosability recently introduced by Mohar and Zhu. First, we provide a negative answer to a question of Zhu about circular cliques. We next prove that, for every graph G, cch(G) = O( ch(G) + ln |V(G)| ). We investigate a generalisation of circular choosability, circular f-choosability, when f is a...
Uploaded on: October 11, 2023 -
July 2, 2008 (v1)Conference paper
Art gallery problems have been extensively studied over the last decade and have found different type of applications. Normally the number of sides of a polygon or the general shape of the polygon is used as a measure of the complexity of the problem. In this paper we explore another measure of complexity, namely, the number of guards required...
Uploaded on: December 4, 2022 -
May 2005 (v1)Conference paper
We model a problem related to routing reconfiguration in WDM networks. We establish some similarities and differ- ences with two other known problems: the pathwidth and the pursuit problem. We then present a distributed linear-time algorithm to solve the problem on trees. Last we give the solutions for some classes of graphs, in particular...
Uploaded on: December 3, 2022 -
2008 (v1)Report
In WDM backbone networks, the traffic pattern evolves constantly due to the nature of the demand itself or because of equipment failures leading to reroute affected connections. In this context, requests are routed greedily using available resources without changing the routing of pre-established connections. However, such a policy leads to a...
Uploaded on: December 4, 2022