Showing papers in "Combinatorics, Probability & Computing in 1998"
TL;DR: The size of the giant component in the former case, and the structure of the graph formed by deleting that component is analyzed, which is basically that of a random graph with n′=n−∣C∣ vertices, and with λ′in′ of them of degree i.
Abstract: 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.
876 citations
TL;DR: An idea of Szekely is applied to prove a general upper bound on the number of incidences between a set of m points and a sets of n ‘well-behaved’ curves in the plane.
Abstract: We apply an idea of Szekely to prove a general upper bound on the number of incidences between a set of m points and a set of n ‘well-behaved’ curves in the plane.
187 citations
TL;DR: For any integer k, it is deduced that, unless NP=RP, no polynomial time algorithm for k-Colourability on graphs G of girth g(G)≥log∣G∣/13logk and maximum degree Δ( G)≤6k13 can exist.
Abstract: 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)≥log∣G∣/13logk and maximum degree Δ(G)≤6k13 can exist. We also study several related problems.
99 citations
TL;DR: It is shown 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.
Abstract: It is known that evaluating the Tutte polynomial, T(G; x, y), of a graph, G, is nP-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.
82 citations
TL;DR: It is proved that there exists n0, such that, for every n≥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.
Abstract: We prove that there exists n0, such that, for every n≥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.
79 citations
TL;DR: 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.
Abstract: The cover time, C, for a simple random walk on a realization, GN, of G(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)≥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−e)N log N≤E[C∣GN] ≤(1+e)N log N for any fixed e>0, whereas if f(N)=O(log N), there exists a constant a>0 such that, with probability 1−o(1), E[C∣GN] ≥(1+a)N log N. It is furthermore shown that if f(N) is of higher order than (log N)3 then Var(C∣GN)= o((N log N)2) with probability 1−o(1), so that with probability 1−o(1), the stronger statement that (1−e)N log N≤C≤(1+e)N log N holds.
50 citations
TL;DR: The concept of list total colourings is studied and it is proved that every multigraph of maximum degree 3 is 5-total-choosable and the total choice number of a graph is equal to its total chromatic number.
Abstract: We study the concept of list total colourings and prove that every multigraph of maximum degree 3 is 5-total-choosable. We also show that the total choice number of a graph of maximum degree 2 is equal to its total chromatic number.
49 citations
TL;DR: This work gives a characterization for isoperimetric invariants, including the Cheeger constant and the is operimetric number of a graph, which leads to an isoperIMetric inequality for the Cartesian products of graphs.
Abstract: We give a characterization for isoperimetric invariants, including the Cheeger constant and the isoperimetric number of a graph. This leads to an isoperimetric inequality for the Cartesian products of graphs.
48 citations
TL;DR: The asymptotic behaviour of the number of Eulerian circuits in a complete graph of odd order is determined and exact values for up to 21 nodes are given.
Abstract: 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.
45 citations
TL;DR: Using trivariate generating functions, an explicit expression is obtained for the variance of the number of comparisons, if the authors search for the jth element in a random permutation of n elements, under the assumption that j is proportional to n.
Abstract: We study the number of comparisons in Hoare's Find algorithm. Using trivariate generating functions, we get an explicit expression for the variance of the number of comparisons, if we search for the jth element in a random permutation of n elements. The variance is also asymptotically evaluated under the assumption that j is proportional to n. Similar results for the number of passes (recursive calls) are given, too.
39 citations
TL;DR: It is proved that the sums of the n-subsequences of μ must include a non-null subgroup and this last result reduces to the Erdos–Ginzburg–Ziv theorem for k=2.
Abstract: 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.
TL;DR: The results of Hoeffding and Azuma and McDiarmid give similar results for functions which can be expressed as martingales with a bounded difference property, which means that each individual choice has a bounded effect on the value of the function.
Abstract: Often when analysing randomized algorithms, especially parallel or distributed algorithms, one is called upon to show that some function of many independent choices is tightly concentrated about its expected value. For example, the algorithm might colour the vertices of a given graph with two colours and one would wish to show that, with high probability, very nearly half of all edges are monochromatic.The classic result of Chernoff [3] gives such a large deviation result when the function is a sum of independent indicator random variables. The results of Hoeffding [5] and Azuma [2] give similar results for functions which can be expressed as martingales with a bounded difference property. Roughly speaking, this means that each individual choice has a bounded effect on the value of the function. McDiarmid [9] nicely summarized these results and gave a host of applications. Expressed a little differently, his main result is as follows.
TL;DR: This paper addresses both density and Ramsey-type questions for affine d-cubes, and improves for upper and lower bounds on h(d,r) are given for d>2.
Abstract: A collection H of integers is called an affine d-cube if there exist d+1 positive integers x0,x1,…, xd so that***** Insert equation here *****We address both density and Ramsey-type questions for affine d-cubes. Regarding density results, upper bounds are found for the size of the largest subset of {1,2,…,n} not containing an affine d-cube. In 1892 Hilbert published the first Ramsey-type result for affine d-cubes by showing that, for any positive integers r and d, there exists a least number n=h(d,r) so that, for any r-colouring of {1,2,…,n}, there is a monochromatic affine d-cube. Improvements for upper and lower bounds on h(d,r) are given for d>2.
TL;DR: In this article, the authors studied three quantities that can each be viewed as the time needed for a finite irreducible Markov chain to "forget" where it started: mixing time, the minimum mean length of any stopping rule that yields the same distribution from any starting state, reset time, and expected time between independent samples from the stationary distribution.
Abstract: We study three quantities that can each be viewed as the time needed for a finite irreducible Markov chain to ‘forget’ where it started One of these is the mixing time, the minimum mean length of a stopping rule that yields the stationary distribution from the worst starting state A second is the forget time, the minimum mean length of any stopping rule that yields the same distribution from any starting state The third is the reset time, the minimum expected time between independent samples from the stationary distributionOur main results state that the mixing time of a chain is equal to the mixing time of the time-reversed chain, while the forget time of a chain is equal to the reset time of the reverse chain In particular, the forget time and the reset time of a time-reversible chain are equal Moreover, the mixing time lies between absolute constant multiples of the sum of the forget time and the reset timeWe also derive an explicit formula for the forget time, in terms of the 'access times' introduced in [11] This enables us to relate the forget and reset times to other mixing measures of the chain
TL;DR: A class of completely symmetric simple Venn diagrams for seven sets is constructed and displayed.
Abstract: A class of completely symmetric simple Venn diagrams for seven sets is constructed and displayed.
TL;DR: The bound is best possible that the number of sums of the subsets of S is at least min(∣G∣, 2∣S∣).
Abstract: Let S be a generating subset of a cyclic group G such that 0=∉S and ∣S∣≥5. We show that the number of sums of the subsets of S is at least min(∣G∣, 2∣S∣). Our bound is best possible. We obtain similar results for abelian groups and mention the generalization to nonabelian groups.
TL;DR: Every sequence of 2n−¼k2+k−2 elements in G assuming k distinct values has an n-subsequence with sum zero, which settles a conjecture of Bialostocki and Lotspeich.
Abstract: Let k be a positive integer and G a finite abelian group of order n, where n≥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.
TL;DR: A finite version of de Finetti's theorem without the exchangeability hypothesis is obtained: for any fixed m, k and e, every sufficiently long sequence of such random variables has a length-k subsequence at variation distance less than e from an i.i.d. mix.
Abstract: We consider probability spaces which contain a family {EArA⊆{1, 2, …, n}, ∣A∣=k} of events indexed by the k-element subsets of {1, 2, …, n}. A pair (A, B) of k-element subsets of {1, 2, …, n} is called a shift pair if the largest k−1 elements of A coincide with the smallest k−1 elements of B. For a shift pair (A, B), Pr[AB¯] is the probability that event EA is true and EB is false. We investigate how large the minimum value of Pr[AB¯], taken over all shift pairs, can be. As n→∞, this value converges to a number λk, with ½−1/2k+2≤λk≤ ½−1/4k+2. We show that λk is a strictly increasing function of k, with λ1=¼ and λ2=1/3.For k=1, our results have the following natural interpretation. If a fair coin is tossed repeatedly, and event Ei is true when the ith toss is heads, then for all i and j with i 0, there is an n such that for any sequence E1, E2, …, En of events in an arbitrary probability space, there are indices i
TL;DR: B Bollobas proved that any real linear combination of the form ∑apkp(G) attains its maximum on a complete multipartite graph and it is shown that the same is true for a linear combinationof the form +bpkp (G¯), provided bp≥0 for every p.
Abstract: Let kp(G) denote the number of complete subgraphs of order p in the graph G. Bollobas proved that any real linear combination of the form ∑apkp(G) attains its maximum on a complete multipartite graph. We show that the same is true for a linear combination of the form ∑apkp(G) +bpkp(G¯), provided bp≥0 for every p.
TL;DR: This work considers the problem of fault diagnosis in multiprocessor systems and investigates 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.
Abstract: We consider the problem of fault diagnosis in multiprocessor systems. Processors perform tests 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 constant probability p<1/2 and the goal is to identify correctly the status of all processors, based on the set of test results. For 0
TL;DR: Stacks which allow elements to be pushed into any of the top r positions and popped from the top s positions are studied and an asymptotic formula for the number un of permutations of length n sortable by such a stack is found.
Abstract: 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.
TL;DR: It is deduced that the maximal zero-free intervals for characteristic polynomials of loopless matroids are precisely (−∞, 1) and (1, 32/27].
Abstract: Let M be a loopless matroid with rank r and c components. Let P(M, t) be the characteristic polynomial of M. We shall show that (−1)rP(M, t)≥(1−t)r for t∈(−∞, 1), that the multiplicity of the zeros of P(M, t) at t=1 is equal to c, and that (−1)r+cP(M, t)≥(t−1)r for t∈(1, 32/27]. Using a result of C. Thomassen we deduce that the maximal zero-free intervals for characteristic polynomials of loopless matroids are precisely (−∞, 1) and (1, 32/27].
TL;DR: A general proof of Welsh's problem about mean first-passage times is given, which, although seemingly simple, has not been proved in any non-trivial case.
Abstract: Consider first-passage percolation on the square lattice. Welsh, who together with Hammersley introduced the subject in 1963, has formulated a problem about mean first-passage times, which, although seemingly simple, has not been proved in any non-trivial case. In this paper we give a general proof of Welsh's problem.
TL;DR: A game in which two players alternately claim one and q edges, respectively, and the game TBin(N, n, q) can be won by the Maker provided that n=o(N).
Abstract: 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 0, there exists n0 such that, for every n≥n0, the Breaker has a winning strategy in TBin(N, n, q) if q>(1+e)N/logn, while, for q<(1−e)N/logn, the game TBin(N, n, q) can be won by the Maker provided that n=o(N).
TL;DR: In this paper, the authors derived precise asymptotic estimates for N(n, m), as n→∞ and m varies through all possible values (in general 1≤m≤n).
Abstract: Given a class of combinatorial structures C, we consider the quantity N(n, m), the number of multiset constructions P (of C) of size n having exactly m C-components. Under general analytic conditions on the generating function of C, we derive precise asymptotic estimates for N(n, m), as n→∞ and m varies through all possible values (in general 1≤m≤n). In particular, we show that the number of C-components in a random (assuming a uniform probability measure) P-structure of size n obeys asymptotically a convolution law of the Poisson and the geometric distributions. Applications of the results include random mapping patterns, polynomials in finite fields, parameters in additive arithmetical semigroups, etc. This work develops the ‘additive’ counterpart of our previous work on the distribution of the number of prime factors of an integer [20].
TL;DR: This result is best possible for nontrivial cross-intersecting families with integers with 2≤a≤b and n≥a+b.
Abstract: Let a, b and n be integers with 2≤a≤b and n≥a+b. Suppose that A⊂([n]a) and B⊂([n]b) are nontrivial cross-intersecting families. Then ∣A∣+∣B∣≤2+(nb) −2(n−ab)+ (n−2ab). This result is best possible.
TL;DR: It is proved that C meets at least two vertex-triads of M, which gives direct binary matroid generalizations of certain graph results of Halin, Lemos, and Mader and generalizes Tutte's Triangle Lemma for general matroids.
Abstract: A cocircuit C* in a matroid M is said to be non-separating if and only if M∖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 ∣E(M)∣≥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.
TL;DR: It is proved that if n≥3k−1, then ak(T)=rk(T), and it is shown that this bound on n is best possible.
Abstract: Let T be a semicomplete digraph on n vertices. Let ak(T) denote the minimum number of arcs whose addition to T results in a k-connected semicomplete digraph and let rk(T) denote the minimum number of arcs whose reversal in T results in a k-connected semicomplete digraph. We prove that if n≥3k−1, then ak(T)=rk(T). We also show that this bound on n is best possible.
TL;DR: It is shown that the random walk starting at a, x is more likely to have hit (a, x) than (b, y) by time t, which contrasts with the discrete time case and proves a conjecture of Bollobas and Brightwell.
Abstract: 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 Bollobas 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.
TL;DR: The construction of a probabilistic distribution on formulas in the basis of∧, [oplus ] 1 in some given set of n variables and of size at most [lscr ](k)=4k is a step towards developing a model appropriate for investigation of the properties of a typical (random) Boolean function of some given complexity.
Abstract: For any Boolean function f, let L(f) be its formula size complexity in the basis {∧, ⊕ 1}. For every n and every k≤n/2, we describe a probabilistic distribution on formulas in the basis {∧, ⊕ 1} in some given set of n variables and of size at most l(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)≤l(k)α, where α=log4(3/2), we have pn,k(f)>0. Moreover, for every function f, if pn,k(f)>0, then***** Insert equation here *****where c>1 is an absolute constant. Although the upper and lower bounds are exponentially small in l(k), they are quasi-polynomially related whenever l(k)≥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.