The tensor product (G1,G2) of a graph G1 and a pointed graph G2 (containing one distinguished edge) is obtained by identifying each edge of G1 with the distinguished edge of a separate copy of G2, and then removing the identified edges. A formula to compute the Tutte polynomial of a tensor product of graphs was originally given by Brylawski. This formula was recently generalized to coloured graphs and the generalized Tutte polynomial introduced by Bollobás and Riordan. In this paper we generalize the coloured tensor product formula to relative Tutte polynomials of relative graphs, containing zero edges to which the usual deletion/contraction rules do not apply. As we have shown in a recent paper, relative Tutte polynomials may be used to compute the Jones polynomial of a virtual knot.

Given a (multi)digraph H, a digraph D is H-linked if every injective function ι:V(H) → V(D) can be extended to an H-subdivision. In this paper, we give sharp degree conditions that ensure a sufficiently large digraph D is H-linked for arbitrary H. The notion of an H-linked digraph extends the classes of m-linked, m-ordered and strongly m-connected digraphs.

First, we give sharp minimum semi-degree conditions for H-linkedness, extending results of Kühn and Osthus on m-linked and m-ordered digraphs. It is known that the minimum degree threshold for an undirected graph to be H-linked depends on a partition of the (undirected) graph H into three parts. Here, we show that the corresponding semi-degree threshold for H-linked digraphs depends on a partition of H into as many as nine parts.

We also determine sharp Ore–Woodall-type degree-sum conditions ensuring that a digraph D is H-linked for general H. As a corollary, we obtain (previously undetermined) sharp degree-sum conditions for m-linked and m-ordered digraphs.

Let Δ ≥ 2 be a fixed integer. We show that the random graph ${\mathcal{G}_{n,p}}$ with $p\gg (\log n/n)^{1/\Delta}$ is robust with respect to the containment of almost spanning bipartite graphs H with maximum degree Δ and sublinear bandwidth in the following sense: asymptotically almost surely, if an adversary deletes arbitrary edges from ${\mathcal{G}_{n,p}}$ in such a way that each vertex loses less than half of its neighbours, then the resulting graph still contains a copy of all such H.

We study cross-graph charging schemes for graphs drawn in the plane. These are charging schemes where charge is moved across vertices of different graphs. Such methods have recently been used to obtain various properties of triangulations that are embedded in a fixed set of points in the plane. We generalize this method to obtain results for various other types of graphs that are embedded in the plane. Specifically, we obtain a new bound of O*(187.53N) (where the O*(⋅) notation hides polynomial factors) for the maximum number of crossing-free straight-edge graphs that can be embedded in any specific set of N points in the plane (improving upon the previous best upper bound 207.85N in Hoffmann, Schulz, Sharir, Sheffer, Tóth and Welzl [14]). We also derive upper bounds for numbers of several other types of plane graphs (such as connected and bi-connected plane graphs), and obtain various bounds on the expected vertex-degrees in graphs that are uniformly chosen from the set of all crossing-free straight-edge graphs that can be embedded in a specific point set.

We then apply the cross-graph charging-scheme method to graphs that allow certain types of crossings. Specifically, we consider graphs with no set of k pairwise crossing edges (more commonly known as k-quasi-planar graphs). For k=3 and k=4, we prove that, for any set S of N points in the plane, the number of graphs that have a straight-edge k-quasi-planar embedding over S is only exponential in N.

Let G be a finite graph with minimum degree r. Form a random subgraph Gp of G by taking each edge of G into Gp independently and with probability p. We prove that for any constant ε > 0, if $p=\frac{1+\epsilon}{r}$, then Gp is non-planar with probability approaching 1 as r grows. This generalizes classical results on planarity of binomial random graphs.

The mean weight of a cycle in an edge-weighted graph is the sum of the cycle's edge weights divided by the cycle's length. We study the minimum mean-weight cycle on the complete graph on n vertices, with random i.i.d. edge weights drawn from an exponential distribution with mean 1. We show that the probability of the min mean weight being at most c/n tends to a limiting function of c which is analytic for c ≤ 1/e, discontinuous at c = 1/e, and equal to 1 for c > 1/e. We further show that if the min mean weight is ≤ 1/(en), then the length of the relevant cycle is Θp(1) (i.e., it has a limiting probability distribution which does not scale with n), but that if the min mean weight is > 1/(en), then the relevant cycle almost always has mean weight (1 + o(1))/(en) and length at least (2/π2 − o (1)) log2n log log n.

Let $\mathcal{H}$ be a set of connected graphs. A graph G is said to be $\mathcal{H}$-free if G does not contain any element of $\mathcal{H}$ as an induced subgraph. Let $\mathcal{F}_{k}(\mathcal{H})$ be the set of k-connected $\mathcal{H}$-free graphs. When we study the relationship between forbidden subgraphs and a certain graph property, we often allow a finite exceptional set of graphs. But if the symmetric difference of $\mathcal{F}_{k}(\mathcal{H}_{1})$ and $\mathcal{F}_{k}(\mathcal{H}_{2})$ is finite and we allow a finite number of exceptions, no graph property can distinguish them. Motivated by this observation, we study when we obtain a finite symmetric difference. In this paper, our main aim is the following. If $|\mathcal{H}|\leq 3$ and the symmetric difference of $\mathcal{F}_{1}(\{H\})$ and $\mathcal{F}_{1}(\mathcal{H})$ is finite, then either $H\in \mathcal{H}$ or $|\mathcal{H}|=3$ and H=C3. Furthermore, we prove that if the symmetric difference of $\mathcal{F}_{k}(\{H_{1}\})$ and $\mathcal{F}_{k}(\{H_{2}\})$ is finite, then H1=H2.

A perfect matching M in an edge-coloured complete bipartite graph Kn,n is rainbow if no pair of edges in M have the same colour. We obtain asymptotic enumeration results for the number of rainbow perfect matchings in terms of the maximum number of occurrences of each colour. We also consider two natural models of random edge-colourings of Kn,n and show that if the number of colours is at least n, then there is with high probability a rainbow perfect matching. This in particular shows that almost every square matrix of order n in which every entry appears n times has a Latin transversal.

A classical result of Robertson and Seymour states that the set of graphs containing a fixed planar graph H as a minor has the so-called Erdős–Pósa property; namely, there exists a function f depending only on H such that, for every graph G and every positive integer k, the graph G has k vertex-disjoint subgraphs each containing H as a minor, or there exists a subset X of vertices of G with |X| ≤ f(k) such that G − X has no H-minor (see Robertson and Seymour, J. Combin. Theory Ser. B41 (1986) 92–114). While the best function f currently known is exponential in k, a O(k log k) bound is known in the special case where H is a forest. This is a consequence of a theorem of Bienstock, Robertson, Seymour and Thomas on the pathwidth of graphs with an excluded forest-minor. In this paper we show that the function f can be taken to be linear when H is a forest. This is best possible in the sense that no linear bound is possible if H has a cycle.

A property of finite graphs is called non-deterministically testable if it has a ‘certificate’ such that once the certificate is specified, its correctness can be verified by random local testing. In this paper we study certificates that consist of one or more unary and/or binary relations on the nodes, in the case of dense graphs. Using the theory of graph limits, we prove that non-deterministically testable properties are also deterministically testable.

The traveling salesman problem (TSP) is one of the most fundamental optimizationproblems. We consider the β-metric traveling salesman problem(Δβ-TSP), i.e., the TSPrestricted to graphs satisfying the β-triangle inequalityc({v,w}) ≤ β(c({v,u}) + c({u,w})),for some cost function c and any three vertices u,v,w.The well-known path matching Christofides algorithm (PMCA) guarantees an approximationratio of 3β2/2 and is the best known algorithm for theΔβ-TSP, for 1 ≤ β ≤ 2. Weprovide a complete analysis of the algorithm. First, we correct an error in the originalimplementation that may produce an invalid solution. Using a worst-case example, we thenshow that the algorithm cannot guarantee a better approximation ratio. The example canalso be used for the PMCA variants for the Hamiltonian path problem with zero and oneprespecified endpoints. For two prespecified endpoints, we cannot reuse the example, butwe construct another worst-case example to show the optimality of the analysis also inthis case.

We provide an algorithm for listing all minimal 2-dominating sets of a tree of ordern in time 𝒪(1.3248n). This implies that every tree has at most1.3248n minimal 2-dominating sets. We also show that thisbound is tight.

We discuss how much space is sufficient to decide whether a unary given numbern is a prime. We show thatO(log log n) space is sufficient for a deterministicTuring machine, if it is equipped with an additional pebble movable along the input tape,and also for an alternating machine, if the space restriction applies only to itsaccepting computation subtrees. In other words, the language is a prime is inpebble–DSPACE(log log n) and also inaccept–ASPACE(log log n). Moreover, if the givenn is composite, such machines are able to find a divisor ofn. Since O(log log n) space is toosmall to write down a divisor, which might requireΩ(log n) bits, the witness divisor is indicated by theinput head position at the moment when the machine halts.