Colouring stability two unit disk graphs

We prove that every stability two unit disk graph has chromatic number at most 3/2 times its clique number

The t-improper chromatic number of random graphs

We consider the $t$-improper chromatic number of the Erd{\H o}s-R{\'e}nyi random graph $G(n,p)$. The t-improper chromatic number $\chi^t(G)$ of $G$ is the smallest number of colours needed in a colouring of the vertices in which each colour class induces a subgraph of maximum degree at most $t$. If $t = 0$, then this is the usual notion of proper colouring. When the edge probability $p$ is constant, we provide a detailed description of the asymptotic behaviour of $\chi^t(G(n,p))$ over the range of choices for the growth of $t = t(n)$.Comment: 12 page

10211 Abstracts Collection -- Flexible Network Design

From Monday 24.05.2010---Friday 28.05.2010, the Dagstuhl Seminar 10211 ``Flexible Network Design \u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Arkhipov's theorem, graph minors, and linear system nonlocal games

The perfect quantum strategies of a linear system game correspond to certain representations of its solution group. We study the solution groups of graph incidence games, which are linear system games in which the underlying linear system is the incidence system of a (non-properly) two-coloured graph. While it is undecidable to determine whether a general linear system game has a perfect quantum strategy, for graph incidence games this problem is solved by Arkhipov's theorem, which states that the graph incidence game of a connected graph has a perfect quantum strategy if and only if it either has a perfect classical strategy, or the graph is nonplanar. Arkhipov's criterion can be rephrased as a forbidden minor condition on connected two-coloured graphs. We extend Arkhipov's theorem by showing that, for graph incidence games of connected two-coloured graphs, every quotient closed property of the solution group has a forbidden minor characterization. We rederive Arkhipov's theorem from the group theoretic point of view, and then find the forbidden minors for two new properties: finiteness and abelianness. Our methods are entirely combinatorial, and finding the forbidden minors for other quotient closed properties seems to be an interesting combinatorial problem.Comment: Minor updates. Also see video abstract at https://youtu.be/uTudADhT1p

WLAN Channel Selection Without Communication

In this paper we consider how a group of wireless access-points can self-configure their channel choice so as to avoid interference between one another and thereby maximise network capacity. We make the observation that communication between access points is not necessary, although it is a feature of almost all published channel allocation algorithms. We argue that this observation is of key practical importance as, except in special circumstances, interfering WLANs need not all lie in the same administrative domain and/or may be beyond wireless communication distance (although within interference distance). We demonstrate the feasibility of the communicationfree paradigm via a new class of decentralized algorithms that are simple, robust and provably correct for arbitrary interference graphs. The algorithm requires only standard hardware and we demonstrate its effectiveness via experimental measurements