Combinatorics, Probability and Computing - Current Issue

Volume 17 Issue 06

Sat, 01 Nov 2008 00:00:00 GMT

Combinatorics, Probability and Computing, Volume 17 Issue 06

Published bimonthly, Combinatorics, Probability & Computing is devoted to the three areas of combinatorics, probability theory and theoretical computer science. Topics covered include classical and algebraic graph theory, extremal set theory, matroid theory, probabilistic methods and random combinatorial structures; combinatorial probability and limit theorems for random combinatorial structures; the theory of algorithms (including complexity theory), randomised algorithms, probabilistic analysis of algorithms, computational learning theory and optimisation.

Coloured Loop-Erased Random Walk on the Complete Graph

Research Articles
JOMY ALAPPATTU, JIM PITMAN,
Combinatorics, Probability and Computing, Volume 17 Issue 06 , pp 727-740

Abstract
Starting from a sequence regarded as a walk through some set of values, we consider the associated loop-erased walk as a sequence of directed edges, with an edge from i to j if the loop-erased walk makes a step from i to j. We introduce a colouring of these edges by painting edges with a fixed colour as long as the walk does not loop back on itself, then switching to a new colour whenever a loop is erased, with each new colour distinct from all previous colours. The pattern of colours along the edges of the loop-erased walk then displays stretches of consecutive steps of the walk left untouched by the loop-erasure process. Assuming that the underlying sequence generating the loop-erased walk is a sequence of independent random variables, each uniform on [N] := {1, 2, . . ., N}, we condition the walk to start at N and stop the walk when it first reaches the subset [k], for some 1 N j k, the events Bi for 1 j .

B 2 [ g ] Sets and a Conjecture of Schinzel and Schmidt

Research Articles
JAVIER CILLERUELO, CARLOS VINUESA,
Combinatorics, Probability and Computing, Volume 17 Issue 06 , pp 741-747

Abstract
A set of integers is called a B2[g] set if every integer m has at most g representations of the form m = a + a a . We obtain a new lower bound for F(g, n), the largest cardinality of a B2[g] set in {1,. . .,n}. More precisely, we prove that infn where g . We show a connection between this problem and another one discussed by Schinzel and Schmidt, which can be considered its continuous version.

Bounds For The Real Zeros of Chromatic Polynomials

Research Articles
F. M. DONG, K. M. KOH,
Combinatorics, Probability and Computing, Volume 17 Issue 06 , pp 749-759

Abstract
Sokal in 2001 proved that the complex zeros of the chromatic polynomial PG(q) of any graph G lie in the disc |q| 7.963907 is the maximum degree of G. This result answered a question posed by Brenti, Royle and Wagner in 1994 and hence proved a conjecture proposed by Biggs, Damerell and Sands in 1972. Borgs gave a short proof of Sokal's result. Fern . In this paper, we shall show that all real zeros of PG(q) are in the interval [0,5.664 = 3, all real zeros of PG(q) are in the interval [0,4.765 ).

Dobrushin Conditions and Systematic Scan

Research Articles
MARTIN DYER, LESLIE ANN GOLDBERG, MARK JERRUM,
Combinatorics, Probability and Computing, Volume 17 Issue 06 , pp 761-779

Abstract
We consider Glauber dynamics on finite spin systems. The mixing time of Glauber dynamics can be bounded in terms of the influences of sites on each other. We consider three parameters bounding these influences: 1 implies that systematic-scan Glauber dynamics (in which sites are updated in a deterministic order) is rapidly mixing. This paper studies two related issues, primarily in the context of systematic scan: (1) the relationship between the parameters Shlosman condition 1 implies rapid mixing of systematic scan. An interesting question is whether the rapid mixing results for scan can be extended to the = 1 cases. As an application, we show rapid mixing of systematic scan (for any scan order) for heat-bath Glauber dynamics for proper q-colourings of a degree- 2 .

Winning Fast in Sparse Graph Construction Games

Research Articles
OHAD N. FELDHEIM, MICHAEL KRIVELEVICH,
Combinatorics, Probability and Computing, Volume 17 Issue 06 , pp 781-791

Abstract
A graph construction game is a Maker Breaker game. Maker and Breaker take turns in choosing previously unoccupied edges of the complete graph KN. Maker's aim is to claim a copy of a given target graph G while Breaker's aim is to prevent Maker from doing so. In this paper we show that if G is a d-degenerate graph on n vertices and N d1122d+9n, then Maker can claim a copy of G in at most d1122d+7n rounds. We also discuss a lower bound on the number of rounds Maker needs to win, and the gap between these bounds.

A Weighted Generalization of Gao's n + D − 1 Theorem

Research Articles
YAHYA O. HAMIDOUNE,
Combinatorics, Probability and Computing, Volume 17 Issue 06 , pp 793-798

Abstract

Individual Displacements in Hashing with Coalesced Chains

Research Articles
SVANTE JANSON,
Combinatorics, Probability and Computing, Volume 17 Issue 06 , pp 799-814

Abstract
We study the asymptotic distribution of the displacements in hashing with coalesced chains, for both late-insertion and early-insertion. Asymptotic formulas for means and variances follow. The method uses Poissonization and some stochastic calculus.

A Weakening of the Odd Hadwiger's Conjecture

Research Articles
KEN-ICHI KAWARABAYASHI,
Combinatorics, Probability and Computing, Volume 17 Issue 06 , pp 815-821

Abstract
Gerards and Seymour (see [10], p. 115) conjectured that if a graph has no odd complete minor of order l, then it is (l 1)-colourable. This is an analogue of the well-known conjecture of Hadwiger, and in fact, this would immediately imply Hadwiger's conjecture. The current best-known bound for the chromatic number of graphs with no odd complete minor of order l is by the recent result by Geelen, Gerards, Reed, Seymour and Vetta [8], and by Kawarabayashi [12] later, independently. But it seems very hard to improve this bound since this would also improve the current best-known bound for the chromatic number of graphs with no complete minor of order l.

When is an Almost Monochromatic K 4 Guaranteed?

Research Articles
ALEXANDR KOSTOCHKA, DHRUV MUBAYI,
Combinatorics, Probability and Computing, Volume 17 Issue 06 , pp 823-830

Abstract
Suppose that n (log k)ck, where c is a fixed positive constant. We prove that, no matter how the edges of Kn are coloured with k colours, there is a copy of K4 whose edges receive at most two colours. This improves the previous best bound of kc is a fixed positive constant, which follows from results on classical Ramsey numbers.

The Hitting Time for the Height of a Random Recursive Tree

Research Articles
THOMAS M. LEWIS,
Combinatorics, Probability and Computing, Volume 17 Issue 06 , pp 831-835

Abstract
In this paper we provide a simple formula for the expected time for a random recursive tree to grow to a given height.

On a Form of Coordinate Percolation

Research Articles
ELIZABETH R. MOSEMAN, PETER WINKLER,
Combinatorics, Probability and Computing, Volume 17 Issue 06 , pp 837-845

Abstract
Let ai,bi, i = 0, 1, 2, . . . be drawn uniformly and independently from the unit interval, and let t be a fixed real number. Let a site (i, j) be open if ai + bj (t) that there is an infinite path (oriented or not) of open sites, containing the origin. )/2.