Let TBin(N, n, q) be the game on the complete graph KN in which two players, the Breaker and the Maker, alternately claim one and q edges, respectively. The Maker's aim is to build a binary tree on n<N vertices in n−1 turns while the Breaker tries to prevent him from doing so. It is shown that, for every constant ε>0, there exists n0 such that, for every n[ges ]n0, the Breaker has a winning strategy in TBin(N, n, q) if q>(1+ε)N/logn, while, for q<(1−ε)N/logn, the game TBin(N, n, q) can be won by the Maker provided that n=o(N).

To bound the probability of a union of n events from a single set of events, Bonferroni inequalities are sometimes used. There are sharper bounds which are called Sobel–Uppuluri–Galambos inequalities. When two (or more) sets of events are involved, bounds are considered on the probability of intersection of several such unions, one union from each set. We present a method for unified treatment of bivariate lower and upper bounds in this note. The lower bounds obtained are new and at least as good as lower bounds appearing in the literature so far. The upper bounds coincide with existing bivariate Sobel–Uppuluri–Galambos type upper bounds derived by the method of indicator functions. A numerical example is given to illustrate that the new lower bounds can be strictly better than existing ones.

Let f(n, k, l) be the expected length of a longest common subsequence of l sequences of length n over an alphabet of size k. It is known that there are constants γ(l)k such that f(n, k, l)→ γ(l)kn as n→∞, and we show that γ(l)k= Θ(k1/l−1) as k→∞. Bounds for the corresponding constants for the expected length of a shortest common supersequence are also presented.

For any integer k, we prove the existence of a uniquely k-colourable graph of girth at least g on at most k12(g+1) vertices whose maximal degree is at most 5k13. From this we deduce that, unless NP=RP, no polynomial time algorithm for k-Colourability on graphs G of girth g(G)[ges ]log[mid ]G[mid ]/13logk and maximum degree Δ(G)[les ]6k13 can exist. We also study several related problems.

A 2×2 table of nonnegative integers is subdivided additively into n 2×2 subtables of nonnegative integers in an arbitrary way. The first case of Simpson's Paradox (SP) occurs when the determinant of the original table is less than zero, but the determinant of each of the n 2×2 subtables is greater than or equal to zero. The second case of SP occurs when the previous inequalities hold with ‘less than’ and ‘greater than or equal’ replaced by ‘greater than’ and ‘less than or equal’, respectively. For the case n=2, this paper calculates the asymptotic proportion of the subdivisions of the original 2×2 table such that SP occurs. It is shown that this asymptotic proportion is bounded above by 1/12.

We consider continuous time random walks on a product graph G×H, where G is arbitrary and H consists of two vertices x and y linked by an edge. For any t>0 and any a, b∈V(G), we show that the random walk starting at (a, x) is more likely to have hit (b, x) than (b, y) by time t. This contrasts with the discrete time case and proves a conjecture of Bollobás and Brightwell. We also generalize the result to cases where H is either a complete graph on n vertices or a cycle on n vertices.

Let G be an abelian group of order n and let μ be a sequence of elements of G with length 2n−k+1 taking k distinct values. Assuming that no value occurs n−k+3 times, we prove that the sums of the n-subsequences of μ must include a non-null subgroup. As a corollary we show that if G is cyclic then μ has an n-subsequence summing to 0. This last result, conjectured by Bialostocki, reduces to the Erdos–Ginzburg–Ziv theorem for k=2.

We consider the function χ(Gk), defined to be the smallest number of colours that can colour a graph G in such a way that no vertices of distance at most k receive the same colour. In particular we shall look at how small a value this function can take in terms of the order and diameter of G. We get general bounds for this and tight bounds for the cases k=2 and k=3.

We prove that there exists n0, such that, for every n[ges ]n0 and every 2-colouring of the edges of the complete graph Kn, one can find two vertex-disjoint monochromatic cycles of different colours which cover all vertices of Kn.

We determine the asymptotic behaviour of the number of Eulerian circuits in a complete graph of odd order. One corollary of our result is the following. If a maximum random walk, constrained to use each edge at most once, is taken on Kn, then the probability that all the edges are eventually used is asymptotic to e3/4n−½. Some similar results are obtained about Eulerian circuits and spanning trees in random regular tournaments. We also give exact values for up to 21 nodes.

For any Boolean function f, let L(f) be its formula size complexity in the basis {∧, [oplus ] 1}. For every n and every k[les ]n/2, we describe a probabilistic distribution on formulas in the basis {∧, [oplus ] 1} in some given set of n variables and of size at most [lscr ](k)=4k. Let pn,k(f) be the probability that the formula chosen from the distribution computes the function f. For every function f with L(f)[les ][lscr ](k)α, where α=log4(3/2), we have pn,k(f)>0. Moreover, for every function f, if pn,k(f)>0, then

formula here

where c>1 is an absolute constant. Although the upper and lower bounds are exponentially small in [lscr ](k), they are quasi-polynomially related whenever [lscr ](k)[ges ]lnΩ(1)n. The construction is a step towards developing a model appropriate for investigation of the properties of a typical (random) Boolean function of some given complexity.

Subgraph expansions are commonly used in the analysis of reliability measures of a failure-prone graph. We show that these expansions are special cases of a general result on the expected value of a random variable defined on a partially ordered set; when applied to random subgraphs, the general result defines a natural association between graph functions. As applications, we consider several graph invariants that measure the connectivity of a graph: the number of connected vertex sets of size k, the number of components of size k, and the total number of components. The expected values of these invariants on a random subgraph are global performance measures that generalize the ones commonly studied. Explicit results are obtained for trees, cycles, and complete graphs. Graphs which optimize these performance measures over a given class of graphs are studied

A cocircuit C* in a matroid M is said to be non-separating if and only if M[setmn ]C*, the deletion of C* from M, is connected. A vertex-triad in a matroid is a three-element non-separating cocircuit. Non-separating cocircuits in binary matroids correspond to vertices in graphs. Let C be a circuit of a 3-connected binary matroid M such that [mid ]E(M)[mid ][ges ]4 and, for all elements x of C, the deletion of x from M is not 3-connected. We prove that C meets at least two vertex-triads of M. This gives direct binary matroid generalizations of certain graph results of Halin, Lemos, and Mader. For binary matroids, it also generalizes a result of Oxley. We also prove that a minimally 3-connected binary matroid M which has at least four elements has at least ½r*(M)+1 vertex-triads, where r*(M) is the corank of the matroid M. An immediate consequence of this result is the following result of Halin: a minimally 3-connected graph with n vertices has at least 2n+6/5 vertices of degree three. We also generalize Tutte's Triangle Lemma for general matroids.