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.

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.

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 [sum ]i(i−2)λi>0 then the graph a.s. has a giant component, while if [sum ]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 ε, λ′0, λ′1 … such that a.s. the giant component, C, has εn+o(n) vertices, and the structure of the graph remaining after deleting C is basically that of a random graph with n′=n−[mid ]C[mid ] vertices, and with λ′in′ of them of degree i.

Let k be a positive integer and G a finite abelian group of order n, where n[ges ]k2−4k+8. Then every sequence of 2n−¼k2+k−2 elements in G assuming k distinct values has an n-subsequence with sum zero. This settles a conjecture of Bialostocki and Lotspeich.

Stacks which allow elements to be pushed into any of the top r positions and popped from any of the top s positions are studied. An asymptotic formula for the number un of permutations of length n sortable by such a stack is found in the cases r=1 or s=1. This formula is found from the generating function of un. The sortable permutations are characterized if r=1 or s=1 or r=s=2 by a forbidden subsequence condition.

The natural relations for sets are those definable in terms of the emptiness of the subsets corresponding to Boolean combinations of the sets. For pairs of sets, there are just five natural relations of interest, namely, strict inclusion in each direction, disjointness, intersection with the universe being covered, or not. Let N denote {1, 2, …, n} and (N2) denote {(i, j)[mid ]i, j∈N and i<j}. A function μ on (N2) specifies one of these relations for each pair of indices. Then μ is said to be consistent on M⊆N if and only if there exists a collection of sets corresponding to indices in M such that the relations specified by μ hold between each associated pair of the sets. Firstly, it is proved that if μ is consistent on all subsets of N of size three then μ is consistent on N. Secondly, explicit conditions that make μ consistent on a subset of size three are given as generalized transitivity laws. Finally, it is shown that the result concerning binary natural relations can be generalized to r-ary natural relations for arbitrary r[ges ]2.

Let a, b and n be integers with 2[les ]a[les ]b and n[ges ]a+b. Suppose that [Ascr]⊂([n]a) and [Bscr]⊂([n]b) are nontrivial cross-intersecting families. Then [mid ][Ascr][mid ]+[mid ][Bscr][mid ][les ]2+(nb)−2(n−ab)+(n−2ab). This result is best possible.

We consider the problem of fault diagnosis in multiprocessor systems. Processors performtests on one another: fault-free testers correctly identify the fault status of tested processors, while faulty testers can give arbitrary test results. Processors fail independently with constantprobability p<1/2 and the goal is to identify correctly the status of all processors, based on the set of test results. For 0<q<1, q-diagnosis is a fault diagnosis algorithm whose probability of error does not exceed q. We show that the minimum number of tests to perform q-diagnosis for n processors is Θ(n log 1/q) in the nonadaptive case and n+Θ( log 1/q) in the adaptive case. We also investigate q-diagnosis algorithms that minimize the maximum number of tests performed by, and performed on, processors in the system, constructing testing schemes in which each processor is involved in very few tests. Our results demonstrate that the flexibility yielded by adaptive testing permits a significant saving in the number of tests for the same reliability of diagnosis.

It is known that evaluating the Tutte polynomial, T(G; x, y), of a graph, G, is #P-hard at all but eight specific points and one specific curve of the (x, y)-plane. In contrast we show that if k is a fixed constant then for graphs of tree-width at most k there is an algorithm that will evaluate the polynomial at any point using only a linear number of multiplications and additions.

The cover time, C, for a simple random walk on a realization, GN, of [Gscr](N, p), the random graph on N vertices where each two vertices have an edge between them with probability p independently, is studied. The parameter p is allowed to decrease with N and p is written on the form f(N)/N, where it is assumed that f(N)[ges ]c log N for some c>1 to asymptotically ensure connectedness of the graph. It is shown that if f(N) is of higher order than log N, then, with probability 1−o(1), (1−ε)N log N[les ]E[C[mid ]GN][les ](1+ε)N log N for any fixed ε>0, whereas if f(N)=O(log N), there exists a constant a>0 such that, with probability 1−o(1), E[C[mid ]GN][ges ](1+a)N log N. It is furthermore shown that if f(N) is of higher order than (log N)3 then Var(C[mid ]GN)=o((N log N)2) with probability 1−o(1), so that with probability 1−o(1), the stronger statement that (1−ε)N log N[les ]C[les ](1+ε)N log N holds.