Proceedings of the International Congress of Mathematicians 2010 (ICM 2010), 2011
Page 1. SUBGRAPHS OF RANDOM GRAPHS WITH SPECIFIED DEGREES Brendan D. McKay Computer Science Depar... more Page 1. SUBGRAPHS OF RANDOM GRAPHS WITH SPECIFIED DEGREES Brendan D. McKay Computer Science Department, Vanderbilt University, Nashville, Tennessee 37235 Let g be a graphical degree sequence and ...
It was conjectured by Paul Erdős that if G is a graph with chromatic number at least k, then the ... more It was conjectured by Paul Erdős that if G is a graph with chromatic number at least k, then the diagonal Ramsey number r(G) ≥ r(K k ). That is, the complete graph K k has the smallest diagonal Ramsey number among the graphs of chromatic number k. This conjecture is shown to be false for k = 4 by verifying that r(W 6 ) = 17, where W 6 is the wheel with 6 vertices, since it is well known that r(K 4 ) = 18. Computational techniques are used to determine r(W 6 ) as well as the Ramsey numbers for other pairs of small order wheels.
We enumerate the nonisomorphic and the distinct one-factorizations of K 12 . We also describe the... more We enumerate the nonisomorphic and the distinct one-factorizations of K 12 . We also describe the algorithm used to obtain the result, and the methods we used to verify these numbers.
Combinatorics Probability and Computing, Dec 1, 1998
We determine the asymptotic behaviour of the number of eulerian circuits in a complete graph of o... more 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 K n , then the probability that all the edges are eventually used is asymptotic to e 3/4 n −1/2 . 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.
Some reference citations in the text of this paper are incorrect and should be amended as listed ... more Some reference citations in the text of this paper are incorrect and should be amended as listed below. The reference list is correct as published. Page 1579, line 5: [23, 34, 38, 39] should read [23, 34, 35, 36] Page 1579, line 9: [40] should read [37] Page 1580, line 8: [31, 33, 35] should read [31, 33, 38] Page 1580, line 14: [36] should read [39] Page 1580, line 36: [36] should read [39] Page 1580, line 38: [37] should read [40] Page 1580, line 42: [38] should read [35] Page 1581, line 3 of caption: [33, 35] should read [33, 38] Page 1583, line 12: [41] should read [42] Page 1587, line 9: [39] should read [36] Page 1587, line 11: [39] should read [36] Page 1587, line 18: [38, 39] should read [35, 36] Page 1587, line 37: [39] should read [36] Page 1589, line 18: [39, 46, 47] should read [36, 46, 47] Page 1590, line 2: [39] should read [36] age 1590, line 3: [39] should read [36]
Let m,n be positive integers. Define T(m,n) to be the transportation polytope consisting of the m... more Let m,n be positive integers. Define T(m,n) to be the transportation polytope consisting of the m x n non-negative real matrices whose rows each sum to 1 and whose columns each sum to m/n. The special case B(n)=T(n,n) is the much-studied Birkhoff-von Neumann polytope of doubly-stochastic matrices. Using a recent asymptotic enumeration of non-negative integer matrices (Canfield and McKay, 2007), we determine the asymptotic volume of T(m,n) as n goes to infinity, with m=m(n) such that m/n neither decreases nor increases too quickly. In particular, we give an asymptotic formula for the volume of B(n).
A graph is hypohamiltonian if it is not Hamiltonian, but the deletion of any single vertex gives ... more A graph is hypohamiltonian if it is not Hamiltonian, but the deletion of any single vertex gives a Hamiltonian graph. Until now, the smallest known planar hypohamiltonian graph had 42 vertices, a result due to Araya and Wiener. That result is here improved upon by 25 planar hypohamiltonian graphs of order 40, which are found through computer-aided generation of certain families of planar graphs with girth 4 and a fixed number of 4-faces. It is further shown that planar hypohamiltonian graphs exist for all orders greater than or equal to 42. If Hamiltonian cycles are replaced by Hamiltonian paths throughout the definition of hypohamiltonian graphs, we get the definition of hypotraceable graphs. It is shown that there is a planar hypotraceable graph of order 154 and of all orders greater than or equal to 156. We also show that the smallest hypohamiltonian planar graph of girth 5 has 45 vertices.
The main contribution of this paper is an asymptotic estimate of $N(n,k)$ which holds if $k$ incr... more The main contribution of this paper is an asymptotic estimate of $N(n,k)$ which holds if $k$ increases with $n$ within generous limits and specialises to functions with a given weight, including the resilient functions. In the case of $k=1$, our estimates are valid for all weights.
Let \svec = (s_1,...,s_m) and \tvec = (t_1,...,t_n) be vectors of nonnegative integer-valued func... more Let \svec = (s_1,...,s_m) and \tvec = (t_1,...,t_n) be vectors of nonnegative integer-valued functions of m,n with equal sum S = sum_{i=1}^m s_i = sum_{j=1}^n t_j. Let M(\svec,\tvec) be the number of m*n matrices with nonnegative integer entries such that the i-th row has row sum s_i and the j-th column has column sum t_j for all i,j. Such matrices occur in many different settings, an important example being the contingency tables (also called frequency tables) important in statistics. Define s=max_i s_i and t=max_j t_j. Previous work has established the asymptotic value of M(\svec,\tvec) as m,n\to\infty with s and t bounded (various authors independently, 1971-1974), and when \svec,\tvec are constant vectors with m/n,n/m,s/n >= c/log n for sufficiently large (Canfield and McKay, 2007). In this paper we extend the sparse range to the case st=o(S^(2/3)). The proof in part follows a previous asymptotic enumeration of 0-1 matrices under the same conditions (Greenhill, McKay and Wang, 2006). We also generalise the enumeration to matrices over any subset of the nonnegative integers that includes 0 and 1.
Let Z m be the ring of integers modulo m. The m-rank of an integer matrix is the largest order of... more Let Z m be the ring of integers modulo m. The m-rank of an integer matrix is the largest order of a square submatrix whose determinant is not divisible by m. We determine the probability that a random rectangular matrix over Z m has a specified m-rank and, if it is square, a specified determinant. These results were previously known only for prime m.
). This paper extends the analytic methods used by the latter paper to obtain an asymptotic formu... more ). This paper extends the analytic methods used by the latter paper to obtain an asymptotic formula for B(S,T,X) in the dense regime where the entries of S and T can vary within certain limits and the row and column sums of X are not too large. As applications, we find the asymptotic number of simple digraphs with given vectors of in-degree and out-degree, and the expected permanent of a {0,1}-matrix with given row and column sums, with both results holding in the dense regime.
... almost certain we also saw לאמכורק and לעמכורק, but after a long day in the Israeli National ... more ... almost certain we also saw לאמכורק and לעמכורק, but after a long day in the Israeli National Library in Giv'at-Ram, the stairs leading to the photocopy machines seem very steep and the pencil ... We remove Rabbi II-4, Rabbi Avraham Saba and Rabbi II-8, Rabbi David Ha-Nagid. ...
Proceedings of the International Congress of Mathematicians 2010 (ICM 2010), 2011
Page 1. SUBGRAPHS OF RANDOM GRAPHS WITH SPECIFIED DEGREES Brendan D. McKay Computer Science Depar... more Page 1. SUBGRAPHS OF RANDOM GRAPHS WITH SPECIFIED DEGREES Brendan D. McKay Computer Science Department, Vanderbilt University, Nashville, Tennessee 37235 Let g be a graphical degree sequence and ...
It was conjectured by Paul Erdős that if G is a graph with chromatic number at least k, then the ... more It was conjectured by Paul Erdős that if G is a graph with chromatic number at least k, then the diagonal Ramsey number r(G) ≥ r(K k ). That is, the complete graph K k has the smallest diagonal Ramsey number among the graphs of chromatic number k. This conjecture is shown to be false for k = 4 by verifying that r(W 6 ) = 17, where W 6 is the wheel with 6 vertices, since it is well known that r(K 4 ) = 18. Computational techniques are used to determine r(W 6 ) as well as the Ramsey numbers for other pairs of small order wheels.
We enumerate the nonisomorphic and the distinct one-factorizations of K 12 . We also describe the... more We enumerate the nonisomorphic and the distinct one-factorizations of K 12 . We also describe the algorithm used to obtain the result, and the methods we used to verify these numbers.
Combinatorics Probability and Computing, Dec 1, 1998
We determine the asymptotic behaviour of the number of eulerian circuits in a complete graph of o... more 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 K n , then the probability that all the edges are eventually used is asymptotic to e 3/4 n −1/2 . 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.
Some reference citations in the text of this paper are incorrect and should be amended as listed ... more Some reference citations in the text of this paper are incorrect and should be amended as listed below. The reference list is correct as published. Page 1579, line 5: [23, 34, 38, 39] should read [23, 34, 35, 36] Page 1579, line 9: [40] should read [37] Page 1580, line 8: [31, 33, 35] should read [31, 33, 38] Page 1580, line 14: [36] should read [39] Page 1580, line 36: [36] should read [39] Page 1580, line 38: [37] should read [40] Page 1580, line 42: [38] should read [35] Page 1581, line 3 of caption: [33, 35] should read [33, 38] Page 1583, line 12: [41] should read [42] Page 1587, line 9: [39] should read [36] Page 1587, line 11: [39] should read [36] Page 1587, line 18: [38, 39] should read [35, 36] Page 1587, line 37: [39] should read [36] Page 1589, line 18: [39, 46, 47] should read [36, 46, 47] Page 1590, line 2: [39] should read [36] age 1590, line 3: [39] should read [36]
Let m,n be positive integers. Define T(m,n) to be the transportation polytope consisting of the m... more Let m,n be positive integers. Define T(m,n) to be the transportation polytope consisting of the m x n non-negative real matrices whose rows each sum to 1 and whose columns each sum to m/n. The special case B(n)=T(n,n) is the much-studied Birkhoff-von Neumann polytope of doubly-stochastic matrices. Using a recent asymptotic enumeration of non-negative integer matrices (Canfield and McKay, 2007), we determine the asymptotic volume of T(m,n) as n goes to infinity, with m=m(n) such that m/n neither decreases nor increases too quickly. In particular, we give an asymptotic formula for the volume of B(n).
A graph is hypohamiltonian if it is not Hamiltonian, but the deletion of any single vertex gives ... more A graph is hypohamiltonian if it is not Hamiltonian, but the deletion of any single vertex gives a Hamiltonian graph. Until now, the smallest known planar hypohamiltonian graph had 42 vertices, a result due to Araya and Wiener. That result is here improved upon by 25 planar hypohamiltonian graphs of order 40, which are found through computer-aided generation of certain families of planar graphs with girth 4 and a fixed number of 4-faces. It is further shown that planar hypohamiltonian graphs exist for all orders greater than or equal to 42. If Hamiltonian cycles are replaced by Hamiltonian paths throughout the definition of hypohamiltonian graphs, we get the definition of hypotraceable graphs. It is shown that there is a planar hypotraceable graph of order 154 and of all orders greater than or equal to 156. We also show that the smallest hypohamiltonian planar graph of girth 5 has 45 vertices.
The main contribution of this paper is an asymptotic estimate of $N(n,k)$ which holds if $k$ incr... more The main contribution of this paper is an asymptotic estimate of $N(n,k)$ which holds if $k$ increases with $n$ within generous limits and specialises to functions with a given weight, including the resilient functions. In the case of $k=1$, our estimates are valid for all weights.
Let \svec = (s_1,...,s_m) and \tvec = (t_1,...,t_n) be vectors of nonnegative integer-valued func... more Let \svec = (s_1,...,s_m) and \tvec = (t_1,...,t_n) be vectors of nonnegative integer-valued functions of m,n with equal sum S = sum_{i=1}^m s_i = sum_{j=1}^n t_j. Let M(\svec,\tvec) be the number of m*n matrices with nonnegative integer entries such that the i-th row has row sum s_i and the j-th column has column sum t_j for all i,j. Such matrices occur in many different settings, an important example being the contingency tables (also called frequency tables) important in statistics. Define s=max_i s_i and t=max_j t_j. Previous work has established the asymptotic value of M(\svec,\tvec) as m,n\to\infty with s and t bounded (various authors independently, 1971-1974), and when \svec,\tvec are constant vectors with m/n,n/m,s/n >= c/log n for sufficiently large (Canfield and McKay, 2007). In this paper we extend the sparse range to the case st=o(S^(2/3)). The proof in part follows a previous asymptotic enumeration of 0-1 matrices under the same conditions (Greenhill, McKay and Wang, 2006). We also generalise the enumeration to matrices over any subset of the nonnegative integers that includes 0 and 1.
Let Z m be the ring of integers modulo m. The m-rank of an integer matrix is the largest order of... more Let Z m be the ring of integers modulo m. The m-rank of an integer matrix is the largest order of a square submatrix whose determinant is not divisible by m. We determine the probability that a random rectangular matrix over Z m has a specified m-rank and, if it is square, a specified determinant. These results were previously known only for prime m.
). This paper extends the analytic methods used by the latter paper to obtain an asymptotic formu... more ). This paper extends the analytic methods used by the latter paper to obtain an asymptotic formula for B(S,T,X) in the dense regime where the entries of S and T can vary within certain limits and the row and column sums of X are not too large. As applications, we find the asymptotic number of simple digraphs with given vectors of in-degree and out-degree, and the expected permanent of a {0,1}-matrix with given row and column sums, with both results holding in the dense regime.
... almost certain we also saw לאמכורק and לעמכורק, but after a long day in the Israeli National ... more ... almost certain we also saw לאמכורק and לעמכורק, but after a long day in the Israeli National Library in Giv'at-Ram, the stairs leading to the photocopy machines seem very steep and the pencil ... We remove Rabbi II-4, Rabbi Avraham Saba and Rabbi II-8, Rabbi David Ha-Nagid. ...
Uploads
Papers by Brendan McKay