Random triangle removal

Starting from a complete graph on $n$ vertices, repeatedly delete the edges of a uniformly chosen triangle. This stochastic process terminates once it arrives at a triangle-free graph, and the fundamental question is to estimate the final number of edges (equivalently, the time it takes the process to finish, or how many edge-disjoint triangles are packed via the random greedy algorithm). Bollob\'as and Erd\H{o}s (1990) conjectured that the expected final number of edges has order $n^{3/2}$, motivated by the study of the Ramsey number $R(3,t)$. An upper bound of $o(n^2)$ was shown by Spencer (1995) and independently by R\"odl and Thoma (1996). Several bounds were given for variants and generalizations (e.g., Alon, Kim and Spencer (1997) and Wormald (1999)), while the best known upper bound for the original question of Bollob\'as and Erd\H{o}s was $n^{7/4+o(1)}$ due to Grable (1997). No nontrivial lower bound was available. Here we prove that with high probability the final number of edges in random triangle removal is equal to $n^{3/2+o(1)}$, thus confirming the 3/2 exponent conjectured by Bollob\'as and Erd\H{o}s and matching the predictions of Spencer et al. For the upper bound, for any fixed $\epsilon>0$ we construct a family of $\exp(O(1/\epsilon))$ graphs by gluing $O(1/\epsilon)$ triangles sequentially in a prescribed manner, and dynamically track all homomorphisms from them, rooted at any two vertices, up to the point where $n^{3/2+\epsilon}$ edges remain. A system of martingales establishes concentration for these random variables around their analogous means in a random graph with corresponding edge density, and a key role is played by the self-correcting nature of the process. The lower bound builds on the estimates at that very point to show that the process will typically terminate with at least $n^{3/2-o(1)}$ edges left.Comment: 42 pages, 4 figures. Supercedes arXiv:1108.178

The game chromatic number of random graphs

Given a graph G and an integer k, two players take turns coloring the vertices of G one by one using k colors so that neighboring vertices get different colors. The first player wins iff at the end of the game all the vertices of G are colored. The game chromatic number \chi_g(G) is the minimum k for which the first player has a winning strategy. In this paper we analyze the asymptotic behavior of this parameter for a random graph G_{n,p}. We show that with high probability the game chromatic number of G_{n,p} is at least twice its chromatic number but, up to a multiplicative constant, has the same order of magnitude. We also study the game chromatic number of random bipartite graphs

Relocating empowerment as a management concept for Asia

Management theories, especially those in the area of human resource management, are predominantly Western-centric in origin and in the empirical testing that underpins them. The purpose of this paper is to explore perceptions of one such theory, employee empowerment, in an Asian context. Information gathered from an open ended questionnaire and focus groups provide an in-depth examination of hotel managers' perceptions and practice of empowerment in the workplace. This study provides tentative indicators of significant culturally-driven differences in the understanding and application of employee empowerment (in terms of both research and practice) between Western and Asian contexts. The results of this study indicate that empowerment in Asian cultures relates much more to the individual and his/her merits, in contrast to organizationally-driven empowerment in Western countries. The findings also indicate that empowerment by hotel managers is related to the level of personal trust the manager has in an employee

On thermodynamic and quantum fluctuations of cosmological constant

We discuss from the condensed-matter point of view the recent idea that the Poisson fluctuations of cosmological constant about zero could be a source of the observed dark energy. We argue that the thermodynamic fluctuations of Lambda are much bigger. Since the amplitude of fluctuations is proportional to V^{-1/2}, where V is the volume of the Universe, the present constraint on the cosmological constant provides the lower limit for V, which is much bigger than the volume within the cosmological horizon.Comment: 4 pages, version submitted to JETP Letter

Ramsey games with giants

The classical result in the theory of random graphs, proved by Erdos and Renyi in 1960, concerns the threshold for the appearance of the giant component in the random graph process. We consider a variant of this problem, with a Ramsey flavor. Now, each random edge that arrives in the sequence of rounds must be colored with one of R colors. The goal can be either to create a giant component in every color class, or alternatively, to avoid it in every color. One can analyze the offline or online setting for this problem. In this paper, we consider all these variants and provide nontrivial upper and lower bounds; in certain cases (like online avoidance) the obtained bounds are asymptotically tight.Comment: 29 pages; minor revision