Cooperative colorings of trees and of bipartite graphs

Given a system $(G_1, \ldots ,G_m)$ of graphs on the same vertex set $V$, a cooperative coloring is a choice of vertex sets $I_1, \ldots ,I_m$, such that $I_j$ is independent in $G_j$ and $\bigcup_{j=1}^{m}I_j = V$. For a class $\mathcal{G}$ of graphs, let $m_{\mathcal{G}}(d)$ be the minimal $m$ such that every $m$ graphs from $\mathcal{G}$ with maximum degree $d$ have a cooperative coloring. We prove that $\Omega(\log\log d) \le m_\mathcal{T}(d) \le O(\log d)$ and $\Omega(\log d)\le m_\mathcal{B}(d) \le O(d/\log d)$, where $\mathcal{T}$ is the class of trees and $\mathcal{B}$ is the class of bipartite graphs.Comment: 8 pages, 2 figures, accepted to the Electronic Journal of Combinatorics, corrections suggested by the referees have been incorporate

Cooperative colorings of trees and of bipartite graphs

International audienceGiven a system (G 1 ,. .. , G m) of graphs on the same vertex set V , a cooperative coloring is a choice of vertex sets I 1 ,. .. , I m , such that I j is independent in G j and m j=1 I j = V. For a class G of graphs, let m G (d) be the minimal m such that every m graphs from G with maximum degree d have a cooperative coloring. We prove that Ω(log log d) m T (d) O(log d) and Ω(log d) m B (d)

Cooperative coloring of some graph families

Given a family of graphs $\{G_1,\ldots, G_m\}$ on the vertex set $V$, a cooperative coloring of it is a choice of independent sets $I_i$ in $G_i$ $(1\leq i\leq m)$ such that $\bigcup^m_{i=1}I_i=V$. For a graph class $\mathcal{G}$, let $m_{\mathcal{G}}(d)$ be the minimum $m$ such that every graph family $\{G_1,\ldots,G_m\}$ with $G_j\in\mathcal{G}$ and $\Delta(G_j)\leq d$ for $j\in [m]$, has a cooperative coloring. For $\mathcal{T}$ the class of trees and $\mathcal{W}$ the class of wheels, we get that $m_\mathcal{T}(3)=4$ and $m_\mathcal{W}(4)=5$. Also, we show that $m_{\mathcal{B}_{bc}}(d)=O(\log_2 d)$ and $m_{\mathcal{B}_k}(d)=O\big(\frac{\log d}{\log\log d}\big)$, where $\mathcal{B}_{bc}$ is the class of graphs whose components are balanced complete bipartite graphs, and $\mathcal{B}_k$ is the class of bipartite graphs with one part size at most $k$

Complexity of Computing the Shapley Value in Games with Externalities

We study the complexity of computing the Shapley value in games with externalities. We focus on two representations based on marginal contribution nets (embedded MC-nets and weighted MC-nets). Our results show that while weighted MC-nets are more concise than embedded MC-nets, they have slightly worse computational properties when it comes to computing the Shapley value

Algorithmic Graph Theory

The main focus of this workshop was on mathematical techniques needed for the development of efficient solutions and algorithms for computationally difficult graph problems. The techniques studied at the workshhop included: the probabilistic method and randomized algorithms, approximation and optimization, structured families of graphs and approximation algorithms for large problems. The workshop Algorithmic Graph Theory was attended by 46 participants, many of them being young researchers. In 15 survey talks an overview of recent developments in Algorithmic Graph Theory was given. These talks were supplemented by 10 shorter talks and by two special sessions

Information Inequalities for Joint Distributions, with Interpretations and Applications

Upper and lower bounds are obtained for the joint entropy of a collection of random variables in terms of an arbitrary collection of subset joint entropies. These inequalities generalize Shannon's chain rule for entropy as well as inequalities of Han, Fujishige and Shearer. A duality between the upper and lower bounds for joint entropy is developed. All of these results are shown to be special cases of general, new results for submodular functions-- thus, the inequalities presented constitute a richly structured class of Shannon-type inequalities. The new inequalities are applied to obtain new results in combinatorics, such as bounds on the number of independent sets in an arbitrary graph and the number of zero-error source-channel codes, as well as new determinantal inequalities in matrix theory. A new inequality for relative entropies is also developed, along with interpretations in terms of hypothesis testing. Finally, revealing connections of the results to literature in economics, computer science, and physics are explored.Comment: 15 pages, 1 figure. Originally submitted to the IEEE Transactions on Information Theory in May 2007, the current version incorporates reviewer comments including elimination of an erro

Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization

International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM

Transversal factors and spanning trees

Given a collection of graphs $\mathbf{G}=(G_1, \ldots, G_m)$ with the same vertex set, an $m$-edge graph $H\subset \cup_{i\in [m]}G_i$ is a transversal if there is a bijection $\phi:E(H)\to [m]$ such that $e\in E(G_{\phi(e)})$ for each $e\in E(H)$. We give asymptotically-tight minimum degree conditions for a graph collection on an $n$-vertex set to have a transversal which is a copy of a graph $H$, when $H$ is an $n$-vertex graph which is an $F$-factor or a tree with maximum degree $o(n/\log n)$.Comment: 21 page