The Tutte polynomial $T(G;x,y)$ of a graph evaluates to many interesting combinatorial quantities at various points in the $(x,y)$ plane, including the number of spanning trees, number of forests, number of acyclic orientations, the reliability polynomial, the partition function of the Q-state Potts model of a graph, and the Jones polynomial of an alternating link. The exact computation of $T(G;x,y)$ has been shown by Vertigan and Welsh [8] to be #P-hard at all but a few special points and on two hyperbolae, even in the restricted class of planar bipartite graphs. Attention has therefore been focused on approximation schemes. To date, positive results have been restricted to the upper half plane $y>1$, and most results have relied on a condition of sufficient denseness in the graph. In this paper we present an approach that yields a fully polynomial randomized approximation scheme for $T(G;x,y)$ for $x>1,\ y=1$, and for $T(G;2,0)$, in a class of sparse graphs. This is the first positive result that includes the important point $(2,0)$.

The notion of the influence of a variable on a Boolean function on a product space has attracted much attention in combinatorics, computer science and other fields. Two of the basic papers dealing with this notion are by Kahn, Kalai and Linial (KKL) and Bourgain, Kahn, Kalai, Katznelson and Linial (BKKKL).

In this paper we survey the results in those papers and offer some simpler proofs, corrections, and extensions of the theorems presented there. We present several related open problems.

We consider a random generalized railway defined as a random 3-regular multigraph where some vertices are regarded as switches that only allow traffic between certain pairs of attached edges. It is shown that the probability that the generalized railway is functioning is linear in the proportion of switches. Thus there is no threshold phenomenon for this property.

Let ${\bf C}$ denote the field of complex numbers and $\Omega_n$ the set of $n$th roots of unity. For $t = 0,\ldots,n-1$, define the ideal $\Im(n,t+1) \subset {\bf C}[x_0,\ldots,x_{t}]$ consisting of those polynomials in $t+1$ variables that vanish on distinct $n$th roots of unity; that is, $f \in \Im(n,t+1)$ if and only if $f(\omega_0,\ldots,\omega_{t}) = 0$ for all $(\omega_0,\ldots,\omega_{t}) \in \Omega_n^{t+1}$ satisfying $\omega_i \neq \omega_j$, for $0 \le i < j \le t$.

In this paper we apply Gröbner basis methods to give a Combinatorial Nullstellensatz characterization of the ideal $\Im(n,t+1)$. In particular, if $f \in {\bf C}[x_0,\ldots,x_{t}]$, then we give a necessary and sufficient condition on the coefficients of $f$ for membership in $\Im(n,t+1)$.

The aim of this paper is to prove a Turán-type theorem for random graphs. For $\gamma >0$ and graphs $G$ and $H$, write $G\to_\gamma H$ if any $\gamma$-proportion of the edges of $G$ spans at least one copy of $H$ in $G$. We show that for every graph $H$ and every fixed real $\delta>0$, almost every graph $G$ in the binomial random graph model $\cG(n,q)$, with $q=q(n)\gg((\log n)^4/n)^{1/d(H)}$, satisfies $G\to_{(\chi(H)-2)/(\chi(H)-1)+\delta}H$, where as usual $\chi(H)$ denotes the chromatic number of $H$ and $d(H)$ is the ‘degeneracy number’ of $H$.

Since $K_l$, the complete graph on $l$ vertices, is $l$-chromatic and $(l-1)$-degenerate, we infer that for every $l\geq2$ and every fixed real $\delta>0$, almost every graph $G$ in the binomial random graph model $\cG(n,q)$, with $q=q(n)\gg((\log n)^4/n)^{1/(l-1)}$, satisfies $G\to_{(l-2)/(l-1)+\delta}K_l$.

Mader asked whether every $C_4$-free graph $G$ contains a subdivision of a complete graph whose order is at least linear in the average degree of $G$. We show that there is a subdivision of a complete graph whose order is almost linear. More generally, we prove that every $K_{s,t}$-free graph of average degree $r$ contains a subdivision of a complete graph of order $r^{\frac{1}{2}{+}\frac{1}{2(s-1)}-o(1)}$.

The random assignment problem is to minimize the cost of an assignment in an $n\times n$ matrix of random costs. In this paper we study the problem for some integer-valued cost distributions. We consider both uniform distributions on $1,2,\dots ,m$, for $m=n$ or $n^2$, and random permutations of $1,2,\dots ,n$ for each row, or of $1,2,\dots ,n^2$ for the whole matrix. We find the limit of the expected cost for the ‘$n^2$’ cases, and prove bounds for the ‘$n$’ cases. This is done by simple coupling arguments together with recent results of Aldous for the continuous case. We also present a simulation study of these cases.

As usual, let us write $G_{n,p}$ for a random graph with vertex set $[n]=\{1, 2, \dots , n\}$, in which the edges are chosen independently, with probability $p$. Similarly, $G_{n,m}$ is a random graph on $[n]$ with $m$ edges. For $p=m/({{n}\atop{2}})$, in many respects, the random graphs $G_{n,m}$ and $G_{n,p}$ are practically indistinguishable. (See [4] for an introduction to random graphs.) When in the late 1950s and early 1960s Erdős and Rényi founded the theory of random graphs, one of the most important problems they left open was the determination of the chromatic number of a random graph $G_{n,m}$. The original question concerned the case $m=O(n)$, but when in the 1970s Erdős popularized the problem, he was asking for good estimates on the chromatic number of $G_{n,m}$ for $m \sim n^2/4$ (equivalently, the chromatic number of $G_{{n,1/2}}$).