Michael Molloy

A critical point for random graphs with a given degree sequence

Given a sequence of nonnegative real numbers λ0, λ1… which sum to 1, we consider random graphs having approximately λi n vertices of degree i. Essentially, we show that if Σ i(i - 2)λi > 0, then such graphs almost surely have a giant component, while if Σ i(i -2)λ. < 0, then almost surely all components in such graphs are small. We can apply these results to Gn,p,Gn.M, and other well-known models of random graphs. There are also applications related to the chromatic number of sparse random graphs.

The Size of the Giant Component of a Random Graph with a Given Degree Sequence

Given a sequence of nonnegative real numbers λ0, λ1, … that sum to 1, we consider a random graph having approximately λin vertices of degree i. In [12] the authors essentially show that if ∑i(i−2)λi>0 then the graph a.s. has a giant component, while if ∑i(i−2)λi<0 then a.s. all components in the graph are small. In this paper we analyse the size of the giant component in the former case, and the structure of the graph formed by deleting that component. We determine e, λ′0, λ′1 … such that a.s. the giant component, C, has en+o(n) vertices, and the structure of the graph remaining after deleting C is basically that of a random graph with n′=n−∣C∣ vertices, and with λ′in′ of them of degree i.

A bound on the chromatic number of the square of a planar graph

Wegner conjectured that the chromatic number of the square of any planar graph G with maximum degree Δ ≥ 8 is bounded by χ(G2) ≤ ⌊3/2 Δ⌋ + 1. We prove the bound χ(G2) ≤ ⌈5/3 Δ⌉ + 78. This is asymptotically an improvement on the previously best-known bound. For large values of Δ we give the bound of χ(G2) ≤ ⌈5/3 Δ⌉ + 25. We generalize this result to L(p, q)-labeling of planar graphs, by showing that λqp(G) ≤ q ⌈5/3 Δ⌉ + 18p + 77q - 18. For each of the results, the proof provides a quadratic time algorithm.

Random Constraint Satisfaction: A More Accurate Picture

In the last few years there has been a great amount of interest in Random Constraint Satisfaction Problems, both from an experimental and a theoretical point of view. Quite intriguingly, experimental results with various models for generating random CSP instances suggest that the probability of such problems having a solution exhibits a “threshold–like” behavior. In this spirit, some preliminary theoretical work has been done in analyzing these models asymptotically, i.e., as the number of variables grows. In this paper we prove that, contrary to beliefs based on experimental evidence, the models commonly used for generating random CSP instances do not have an asymptotic threshold. In particular, we prove that asymptotically almost all instances they generate are overconstrained, suffering from trivial, local inconsistencies. To complement this result we present an alternative, single–parameter model for generating random CSP instances and prove that, unlike current models, it exhibits non–trivial asymptotic behavior. Moreover, for this new model we derive explicit bounds for the narrow region within which the probability of having a solution changes dramatically.

Further algorithmic aspects of the local lemma

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A Bound on the Strong Chromatic Index of a Graph

We show that the strong chromatic index of a graph with maximum degree�; is at most (2��)�2, for some�>0. This answers a question of Erdo�s and Ne�et�il.

A sharp threshold in proof complexity

We give the first example of a sharp threshold in proof complexity. More precisely, we show that for any sufficiently small <italic>ε>0</italic> and <italic>Δ>2.28</italic>, random formulas consisting of <italic>(1-ε)n</italic> 2-clauses and <italic>&Dgr n</italic> 3-clauses, which are known to be unsatisfiable almost certainly, almost certainly require resolution and Davis-Putnam proofs of unsatisfiability of exponential size, whereas it is easily seen that random formulas with <italic>(1+ε)n</italic> 2-clauses (and <italic>Δ n</italic> 3 clauses) have linear size proofs of unsatisfiability almost certainly. A consequence of our result also yields the first proof that typical random 3-CNF formulas at ratios below the generally accepted range of the satisfiability threshold (and thus expected to be satisfiable almost certainly) cause natural Davis-Putnam algorithms to take exponential time to find satisfying assignments.

A Bound on the Total Chromatic Number

. The proof is probabilistic.

The analysis of a list-coloring algorithm on a random graph

We introduce a natural k-coloring algorithm and analyze its performance on random graphs with constant expected degree c (G/sub n,p=c/n/). For k=3 our results imply that almost all graphs with n vertices and 1.923 n edges are 3-colorable. This improves the lower bound on the threshold for random 3-colorability significantly and settles the last case of a long-standing open question of Bollobas. We also provide a tight asymptotic analysis of the algorithm. We show that for all k/spl ges/3, if c/spl les/k In k-3/2k then the algorithm almost surely succeeds, while for any /spl epsiv/>0, and k sufficiently large, if c/spl ges/(1+/spl epsiv/)k In k then the algorithm almost surely fails. The analysis is based on the use of differential equations to approximate the mean path of certain Markov chains.

Frequency Channel Assignment on Planar Networks

For integers p ? q, a L(p, q)-labeling of a network G is an integer labeling of the nodes of G such that adjacent nodes receive integers which differ by at least p, and nodes at distance two receive labels which differ by at least q. The minimum number of labels required in such labeling is ?qp(G). This arises in the context of frequency channel assignment in mobile and wireless networks and often G is planar. We show that if G is planar then ?qp(G) ? 5/3 (2q - 1)? + 12p + 144q - 78. We also provide an O(n2) time algorithm to find such a labeling. This provides a (5/3 + o(1))-approximation algorithm for the interesting case of q = 1, improving the best previous approximation ratio of 2.