Answering a question of Bang-Jensen and Thomassen [4], we prove that the minimum feedback arc set problem is NP-hard for tournaments.

A simple first moment argument shows that in a randomly chosen $k$-SAT formula with $m$ clauses over $n$ boolean variables, the fraction of satisfiable clauses is $1-2^{-k}+o(1)$ as $m/n\rightarrow\infty$ almost surely. In this paper, we deal with the corresponding algorithmic strong refutation problem: given a random $k$-SAT formula, can we find a certificate that the fraction of satisfiable clauses is $1-2^{-k}+o(1)$ in polynomial time? We present heuristics based on spectral techniques that in the case $k=3$ and $m\geq\ln(n)^6n^{3/2}$, and in the case $k=4$ and $m\geq Cn^2$, find such certificates almost surely. In addition, we present heuristics for bounding the independence number (resp. the chromatic number) of random $k$-uniform hypergraphs from above (resp. from below) for $k=3,4$.

We study negative dependence properties of a sampling process due to Srinivasan to produce distributions on level sets with given marginals. We give a simple proof that the distribution satisfies negative association. We also show that under a linear match schedule it satisfies the stronger condition of conditional negative association via a non-trivial application of the Feder–Mihail theorem. This method involves the notion of a variable of positive influence. We give some results and related counter-examples which might shed some light on its role in a theory of negative dependence.

We consider the complexity of approximating the partition function of the ferromagnetic Ising model with varying interaction energies and local external magnetic fields. Jerrum and Sinclair provided a fully polynomial randomized approximation scheme for the case in which the system is consistent in the sense that the local external fields all favour the same spin. We characterize the complexity of the general problem by showing that it is equivalent in complexity to the problem of approximately counting independent sets in bipartite graphs, thus it is complete in a logically defined subclass of #P previously studied by Dyer, Goldberg, Greenhill and Jerrum. By contrast, we show that the corresponding computational task for the $q$-state Potts model with local external magnetic fields and $q>2$ is complete for all of #P with respect to approximation-preserving reductions.

A cylinder graph is the graph Cartesian product of a path and a cycle. In this paper we investigate the length of a minimal spanning tree of a cylinder graph whose edges are assigned random lengths according to independent and uniformly distributed random variables. Our work was inspired by a formula of J. Michael Steele which shows that the expected length of a minimal spanning tree of a connected graph can be calculated through the Tutte polynomial of the graph. First, using transfer matrices, we show how to calculate the Tutte polynomials of cylinder graphs. Second, using Steele's formula, we tabulate the expected lengths of the minimal spanning trees for some cylinder graphs. Third, for a fixed cycle length, we show that the ratio of the expected length of a minimal spanning tree of a cylinder graph to the length of the cylinder graph converges to a constant; this constant is described in terms of the Perron–Frobenius eigenvalue of the accompanying transfer matrix. Finally, we show that the length of a minimal spanning tree of a cylinder graph satisfies a strong law of large numbers.

Let $P(G,t)$ and $F(G,t)$ denote the chromatic and flow polynomials of a graph $G$. G. D. Birkhoff and D C. Lewis showed that, if $G$ is a plane near-triangulation, then the only zeros of $P(G,t)$ in $(-\infty,2]$ are 0, 1 and 2. We will extend their theorem by showing that a stronger result to the dual statement holds for both planar and non-planar graphs: if $G$ is a bridge graph with at most one vertex of degree other than three, then the only zeros of $F(G,t)$ in $(-\infty,\alpha]$ are 1 and 2, where $\alpha\approx 2.225\cdots$ is the real zero in $(2,3)$ of the polynomial $t^4-8t^3+22t^2-28t+17$. In addition we construct a sequence of ‘near-cubic’ graphs whose flow polynomials have zeros converging to $\alpha$ from above.

For a fixed graph $H$, we define the rainbow Turán number $\ex^*(n,H)$ to be the maximum number of edges in a graph on $n$ vertices that has a proper edge-colouring with no rainbow $H$. Recall that the (ordinary) Turán number $\ex(n,H)$ is the maximum number of edges in a graph on $n$ vertices that does not contain a copy of $H$. For any non-bipartite $H$ we show that $\ex^*(n,H)=(1+o(1))\ex(n,H)$, and if $H$ is colour-critical we show that $\ex^{*}(n,H)=\ex(n,H)$. When $H$ is the complete bipartite graph $K_{s,t}$ with $s \leq t$ we show $\ex^*(n,K_{s,t}) = O(n^{2-1/s})$, which matches the known bounds for $\ex(n,K_{s,t})$ up to a constant. We also study the rainbow Turán problem for even cycles, and in particular prove the bound $\ex^*(n,C_6) = O(n^{4/3})$, which is of the correct order of magnitude.

This paper is devoted to an online variant of the minimum spanning tree problem in randomly weighted graphs. We assume that the input graph is complete and the edge weights are uniformly distributed over [0,1]. An algorithm receives the edges one by one and has to decide immediately whether to include the current edge into the spanning tree or to reject it. The corresponding edge sequence is determined by some adversary. We propose an algorithm which achieves $\mathbb{E}[ALG]/\mathbb{E}[OPT]=O(1)$ and $\mathbb{E}[ALG/OPT]=O(1)$ against a fair adaptive adversary, i.e., an adversary which determines the edge order online and is fair in a sense that he does not know more about the edge weights than the algorithm. Furthermore, we prove that no online algorithm performs better than $\mathbb{E}[ALG]/\mathbb{E}[OPT]=\Omega(\log n)$ if the adversary knows the edge weights in advance. This lower bound is tight, since there is an algorithm which yields $\mathbb{E}[ALG]/\mathbb{E}[OPT]=O(\log n)$ against the strongest-imaginable adversary.

For an integer $b \geq 1$, the $b$-choice number of a graph $G$ is the minimum integer $k$ such that, for every assignment of a set $S(v)$ of at least $k$ colours to each vertex $v$ of $G$, there is a $b$-set colouring of $G$ that assigns to each vertex $v$ a $b$-set $B(v) \subseteq S(v) \; (|B(v)|=b)$ so that adjacent vertices receive disjoint $b$-sets. This is a generalization of the notions of choice number and chromatic number of a graph. Using probabilistic arguments, we show that, for some positive constant $c > 0$ (independent of $b$), the $b$-choice number of any graph $G$ on $n$ vertices is at most $c (b\chi) (\ln (n/\chi)+1)$ where $\chi = \chi(G)$ denotes the chromatic number of $G$. For any fixed $b$, this bound is tight up to a constant factor for each $n,\chi$. This generalizes and extends a result of Noga Alon [1]wherein a similar bound was obtained for 1-choice numbers of complete $\chi$-partite graphs with each part having size $n/\chi$. We also show that the proof arguments are constructive, leading to polynomial time algorithms for the list colouring problem on certain classes of graphs, provided each vertex is given a list of sufficiently large size.

We answer three questions posed in a paper by Babson and Benjamini. They introduced a parameter $C_G$ for Cayley graphs $G$ that has significant application to percolation. For a minimal cutset of $G$ and a partition of this cutset into two classes, take the minimal distance between the two classes. The supremum of this number over all minimal cutsets and all partitions is $C_G$. We show that if it is finite for some Cayley graph of the group then it is finite for any (finitely generated) Cayley graph. Having an exponential bound for the number of minimal cutsets of size $n$ separating $o$ from infinity also turns out to be independent of the Cayley graph chosen. We show a 1-ended example (the lamplighter group), where $C_G$ is infinite. Finally, we give a new proof for a question of de la Harpe, proving that the number of $n$-element cutsets separating $o$ from infinity is finite unless $G$ is a finite extension of $\mathbb{Z}$.

We say that $n$-vertex graphs $G_1,G_2,\ldots,G_k$pack if there exist injective mappings of their vertex sets onto $[n] = \{1, \ldots,n \}$ such that the images of the edge sets do not intersect. The notion of packing allows one to make some problems on graphs more natural or more general. Clearly, two $n$-vertex graphs $G_1$ and $G_2$ pack if and only if $G_1$ is a subgraph of the complement $\overline{G}_2$ of $G_2$.