A new upper bound on the game chromatic index of graphs

We study the two-player game where Maker and Breaker alternately color the edges of a given graph $G$ with $k$ colors such that adjacent edges never get the same color. Maker's goal is to play such that at the end of the game, all edges are colored. Vice-versa, Breaker wins as soon as there is an uncolored edge where every color is blocked. The game chromatic index $\chi'_g(G)$ denotes the smallest $k$ for which Maker has a winning strategy. The trivial bounds $\Delta(G) \le \chi_g'(G) \le 2\Delta(G)-1$ hold for every graph $G$, where $\Delta(G)$ is the maximum degree of $G$. In 2008, Beveridge, Bohman, Frieze, and Pikhurko proved that for every $\delta>0$ there exists a constant $c>0$ such that $\chi'_g(G) \le (2-c)\Delta(G)$ holds for any graph with $\Delta(G) \ge (\frac{1}{2}+\delta)v(G)$, and conjectured that the same holds for every graph $G$. In this paper, we show that $\chi'_g(G) \le (2-c)\Delta(G)$ is true for all graphs $G$ with $\Delta(G) \ge C \log v(G)$. In addition, we consider a biased version of the game where Breaker is allowed to color $b$ edges per turn and give bounds on the number of colors needed for Maker to win this biased game.Comment: 17 page

A Solution to the 1-2-3 Conjecture

We show that for every graph without isolated edge, the edges can be assigned weights from {1,2,3} so that no two neighbors receive the same sum of incident edge weights. This solves a conjecture of Karo\'{n}ski, Luczak, and Thomason from 2004.Comment: 16 page

Sampling Geometric Inhomogeneous Random Graphs in Linear Time

Real-world networks, like social networks or the internet infrastructure, have structural properties such as large clustering coefficients that can best be described in terms of an underlying geometry. This is why the focus of the literature on theoretical models for real-world networks shifted from classic models without geometry, such as Chung-Lu random graphs, to modern geometry-based models, such as hyperbolic random graphs. With this paper we contribute to the theoretical analysis of these modern, more realistic random graph models. Instead of studying directly hyperbolic random graphs, we use a generalization that we call geometric inhomogeneous random graphs (GIRGs). Since we ignore constant factors in the edge probabilities, GIRGs are technically simpler (specifically, we avoid hyperbolic cosines), while preserving the qualitative behaviour of hyperbolic random graphs, and we suggest to replace hyperbolic random graphs by this new model in future theoretical studies. We prove the following fundamental structural and algorithmic results on GIRGs. (1) As our main contribution we provide a sampling algorithm that generates a random graph from our model in expected linear time, improving the best-known sampling algorithm for hyperbolic random graphs by a substantial factor O(n^0.5). (2) We establish that GIRGs have clustering coefficients in {\Omega}(1), (3) we prove that GIRGs have small separators, i.e., it suffices to delete a sublinear number of edges to break the giant component into two large pieces, and (4) we show how to compress GIRGs using an expected linear number of bits.Comment: 25 page

Economic development, human development, and the pursuit of happiness, April 1, 2, and 3, 2004

This repository item contains a single issue of the Pardee Conference Series, a publication series that began publishing in 2006 by the Boston University Frederick S. Pardee Center for the Study of the Longer-Range Future. This was the Center's spring conference, which took place during April 1, 2, and 3, 2004.The conference asks the questions, how can we make sure that the benefits of economic growth flow into health, education, welfare, and other aspects of human development; and what is the relationship between human development and economic development? Speakers and participants discuss the role that culture, legal and political institutions, the UN Developmental Goals, the level of decision-making, and ethics, play in development

Looking ahead: forecasting and planning for the longer-range future, April 1, 2, and 3, 2005

This repository item contains a single issue of the Pardee Conference Series, a publication series that began publishing in 2006 by the Boston University Frederick S. Pardee Center for the Study of the Longer-Range Future. This was the Center's spring Conference that took place during April 1, 2, and 3, 2005.The conference allowed for many highly esteemed scholars and professionals from a broad range of fields to come together to discuss strategies designed for the 21st century and beyond. The speakers and discussants covered a broad range of subjects including: long-term policy analysis, forecasting for business and investment, the National Intelligence Council Global Trends 2020 report, Europe’s transition from the Marshal plan to the EU, forecasting global transitions, foreign policy planning, and forecasting for defense