Acyclic Coloring of Graphs of Maximum Degree Δ\Delta

International audienceAn acyclic coloring of a graph $G$ is a coloring of its vertices such that: (i) no two neighbors in $G$ are assigned the same color and (ii) no bicolored cycle can exist in $G$. The acyclic chromatic number of $G$ is the least number of colors necessary to acyclically color $G$, and is denoted by $a(G)$. We show that any graph of maximum degree $\Delta$ has acyclic chromatic number at most $\frac{\Delta (\Delta -1) }{ 2}$ for any $\Delta \geq 5$, and we give an $O(n \Delta^2)$ algorithm to acyclically color any graph of maximum degree $\Delta$ with the above mentioned number of colors. This result is roughly two times better than the best general upper bound known so far, yielding $a(G) \leq \Delta (\Delta -1) +2$. By a deeper study of the case $\Delta =5$, we also show that any graph of maximum degree $5$ can be acyclically colored with at most $9$ colors, and give a linear time algorithm to achieve this bound

Minimum feedback vertex set and acyclic coloring

International audienceIn the feedback vertex set problem, the aim is to minimize, in a connected graph G =(V,E), the cardinality of the set overline(V) (G) \subseteq V , whose removal induces an acyclic subgraph. In this paper, we show an interesting relationship between the minimum feedback vertex set problem and the acyclic coloring problem (which consists in coloring vertices of a graph G such that no two colors induce a cycle in G). Then, using results from acyclic coloring, as well as other techniques, we are able to derive new lower and upper bounds on the cardinality of a minimum feedback vertex set in large families of graphs, such as graphs of maximum degree 3, of maximum degree 4, planar graphs, outerplanar graphs, 1-planar graphs, k-trees, etc. Some of these bounds are tight (outerplanar graphs, k-trees), all the others differ by a multiplicative constant never exceeding 2

Minimum feedback vertex set and acyclic coloring

International audienceIn the feedback vertex set problem, the aim is to minimize, in a connected graph G =(V,E), the cardinality of the set overline(V) (G) \subseteq V , whose removal induces an acyclic subgraph. In this paper, we show an interesting relationship between the minimum feedback vertex set problem and the acyclic coloring problem (which consists in coloring vertices of a graph G such that no two colors induce a cycle in G). Then, using results from acyclic coloring, as well as other techniques, we are able to derive new lower and upper bounds on the cardinality of a minimum feedback vertex set in large families of graphs, such as graphs of maximum degree 3, of maximum degree 4, planar graphs, outerplanar graphs, 1-planar graphs, k-trees, etc. Some of these bounds are tight (outerplanar graphs, k-trees), all the others differ by a multiplicative constant never exceeding 2

On the oriented chromatic number of grids

International audienceIn this paper, we focus on the oriented coloring of graphs. Oriented coloring is a coloring of the vertices of an oriented graph $G$ without symmetric arcs such that (i) no two neighbors in $G$ are assigned the same color, and (ii) if two vertices $u$ and $v$ such that $(u,v)\in A(G)$ are assigned colors $c(u)$ and $c(v)$, then for any $(z,t)\in A(G)$, we cannot have simultaneously $c(z)=c(v)$ and $c(t)=c(u)$. The oriented chromatic number of an unoriented graph $G$ is the smallest number $k$ of colors for which any of the orientations of $G$ can be colored with $k$ colors. The main results we obtain in this paper are bounds on the oriented chromatic number of particular families of planar graphs, namely 2-dimensional grids, fat trees and fat fat trees

“Le patrimoine, c’est un truc pour les vieux…”

En dépit des possibilités offertes par le développement culturel et la mise en valeur du patrimoine dans le secteur du tourisme, les politiques publiques de ce type peuvent également faire l’objet de critiques et de contestation sociale. Dans la commune d’Abondance, petite station de ski de Haute-Savoie, la municipalité a tenté en 2007 de fermer définitivement les remontées mécaniques pour développer un tourisme autour du patrimoine gothique de la cour de Savoie. De fortes contestations sont notamment apparues autour des points suivants : 1) le manque de viabilité économique (ainsi que le résume l’un des responsables, “les gens n’y croient pas”) ; 2) à l’inverse du ski en station, le tourisme patrimonial est associé à l’imaginaire négatif de la vieillesse, de l’immobilité, voire de la mort ; 3) pour les résidents de la commune, la mise en place d’un site de visite localisé comporte les inconvénients d’une fréquentation de type turnover.Despite the opportunities offered by cultural heritage in the tourism sector, this article, based on a case study of a little town in the French Alps, shows that public policy of this type may also be subject to criticism and contestation. In Abondance, the municipality is making efforts to convert their ski resort into a cultural tourism site which is built around the heritage of the Gothic Court of Savoy. However: 1) As summarised by one of the local leaders ‘the people do not believe in it’ and fear that is not economically viable. 2) In contrast to the ski resort, heritage tourism is only attached to the imagination of old age, immobility and even death. 3) For residents of the town, setting up a localised visit site has disadvantages because of the turnover of touristic traffic to which this type of tourism is exposed

On star edge colorings of bipartite and subcubic graphs

A star edge coloring of a graph is a proper edge coloring with no $2$-colored path or cycle of length four. The star chromatic index $\chi'_{st}(G)$ of $G$ is the minimum number $t$ for which $G$ has a star edge coloring with $t$ colors. We prove upper bounds for the star chromatic index of complete bipartite graphs; in particular we obtain tight upper bounds for the case when one part has size at most $3$. We also consider bipartite graphs $G$ where all vertices in one part have maximum degree $2$ and all vertices in the other part has maximum degree $b$. Let $k$ be an integer ($k\geq 1$), we prove that if $b=2k+1$ then $\chi'_{st}(G) \leq 3k+2$; and if $b=2k$, then $\chi'_{st}(G) \leq 3k$; both upper bounds are sharp. Finally, we consider the well-known conjecture that subcubic graphs have star chromatic index at most $6$; in particular we settle this conjecture for cubic Halin graphs.Comment: 18 page