We provide an alternate proof of the central limit theorem for the uctuations of the size of the giant component in sparse random graphs. In contrast with previous proofs, the argument investigates a depth-first search algorithm, through first-passage analysis using couplings and martingale limit theorems. The analysis of the first passage limiting distribution for sequences of Markov chains might be interesting in its own right. This proof naturally provides an upper bound for the rate of convergence.

We prove that if an involution ϕ is an automorphism of a connected binary matroid M then there is a hyperplane of M that is invariant under ϕ. We also consider extensions of this result for higher connectivity.

Let F = {G1, …, Gt} be a family of n-vertex graphs defined on the same vertex-set V, and let k be a positive integer. A subset of vertices D ⊂ V is called an (F, k)-core if, for each v ∈ V and for each i = 1, …, t, there are at least k neighbours of v in Gi that belong to D. The subset D is called a connected (F, k)-core if the subgraph induced by D in each Gi is connected. Let δi be the minimum degree of Gi and let δ(F) = minti=1δi. Clearly, an (F, k)-core exists if and only if δ(F) [ges ] k, and a connected (F, k)-core exists if and only if δ(F) [ges ] k and each Gi is connected. Let c(k, F) and cc(k, F) be the minimum size of an (F, k)-core and a connected (F, k)-core, respectively. The following asymptotic results are proved for every t < ln ln δ and k < √lnδ:

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The results are asymptotically tight for infinitely many families F. The results unify and extend related results on dominating sets, strong dominating sets and connected dominating sets.

We consider graphs Gn generated by multisets [Iscr ]n with n random integers as elements, such that vertices of Gn are connected by edges if the elements of [Iscr ]n that the vertices represent are the same, and prove asymptotic results on the sparsity of edges connecting the different subgraphs Gn of the random graph generated by ∪∞n=1[Iscr ]n. These results are of independent interest and, for two models of the bootstrap, we also use them here to link almost sure and complete convergence of the corresponding bootstrap means and averages of related randomly chosen subsequences of a sequence of independent and identically distributed random variables with a finite mean. Complete convergence of these means and averages is then characterized in terms of a relationship between a moment condition on the bootstrapped sequence and the bootstrap sample size. While we also obtain new sufficient conditions for the almost sure convergence of bootstrap means, the approach taken here yields the first necessary conditions.

Let A be an m × n matrix and q := [lfloor ]log2(m)[rfloor ]+1. In this article we improve the well-known bound lindisc(A) [les ] 2 herdisc(A) and show that

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As with the previous proofs relating to this problem, ours is constructive. We will give an on-line algorithm and analyse it using game theory.

In [9, p. 469], Oxley made the following conjecture, which is a geometric analogue of a conjecture of Lovász (see [1, p. 290]) about complete graphs.

Conjecture 1.1.Let G be a rank-n GF(q)-representable simple matroid with critical exponent n − γ. If, for every coline X in G, c(G/X; q) = c(G; q) − 2 = n − γ − 2, then G is the projective geometry PG(n − 1, q).

We shall call the rank n, the critical ‘co-exponent’ γ, and the order q of the field the parameters of Oxley's conjecture. We exhibit several counterexamples to this conjecture. These examples show that, for a given prime power q and a given positive integer γ, Oxley's conjecture holds for only finitely many ranks n. We shall assume familiarity with matroid theory and, in particular, the theory of critical problems. See [6] and [9].

A subset C of points of PG(n − 1, q) is a (γ, k)-cordon if, for every k-codimensional subspace X in PG(n − 1, q), the intersection C ∩ X contains a γ-dimensional subspace of PG(n − 1, q). In this paper, our primary interest will be in constructing (γ, 2)-cordons. With straightforward modifications, our methods will also yield (γ, k)-cordons.

Complements of counterexamples to Oxley's conjecture are (γ, 2)-cordons.

This paper gives a general expression for the Tutte polynomial of perfect matroid designs, that is, for matroids in which all ats of the same rank are equicardinal. In particular, explicit expressions are given for the Tutte polynomial of projective and affine geometries over a finite field.

A question recently posed by Häggkvist and Scott asked whether or not there exists a constant c such that, if G is a graph of minimum degree ck, then G contains cycles of k consecutive even lengths. In this paper we answer the question by proving that, for k > 2, a bipartite graph of average degree at least 4k and girth g contains cycles of (g/2 − 1)k consecutive even lengths. We also obtain a short proof of the theorem of Bondy and Simonovits, that a graph of order n and size at least 8(k − 1)n1+1/k has a cycle of length 2k.

Let G = (V, E) be a graph with n vertices, chromatic number χ(G) and list chromatic number χ[lscr ](G). Suppose each vertex of V(G) is assigned a list of t colours. Albertson, Grossman and Haas [1] conjectured that at least [formula here] vertices can be coloured properly from these lists.

Albertson, Grossman and Haas [1] and Chappell [3] proved partial results concerning this conjecture. This paper presents algorithms that colour at least the number of vertices given in the bounds of Albertson, Grossman and Haas, and Chappell. In particular, it follows that the conjecture is valid for all bipartite graphs and that, for every bipartite graph and every assignment of lists with t colours in each list where 0 [les ] t [les ] χ[lscr ](G), it is possible to colour at least (1 − (1/2)t)n vertices in polynomial time. Thus, if G is bipartite and [Lscr ] is a list assignment with [mid ]L(v)[mid ] [ges ] log2n for all v ∈ V, then G is [Lscr ]-list colourable in polynomial time.