An embedding of a metric graph $(G,d)$ on a closed hyperbolic surface is essential if each complementary region has a negative Euler characteristic. We show, by construction, that given any metric graph, its metric can be rescaled so that it admits an essential and isometric embedding on a closed hyperbolic surface. The essential genus $g_{e}(G)$ of $(G,d)$ is the lowest genus of a surface on which such an embedding is possible. We establish a formula to compute $g_{e}(G)$ and show that, for every integer $g\geq g_{e}(G)$, there is an embedding of $(G,d)$ (possibly after a rescaling of $d$) on a surface of genus $g$. Next, we study minimal embeddings where each complementary region has Euler characteristic $-1$. The maximum essential genus $g_{e}^{\max }(G)$ of $(G,d)$ is the largest genus of a surface on which the graph is minimally embedded. We describe a method for an essential embedding of $(G,d)$, where $g_{e}(G)$ and $g_{e}^{\max }(G)$ are realised.

We consider a problem introduced by Mossel and Ross (‘Shotgun assembly of labeled graphs’, arXiv:1504.07682). Suppose a random n × n jigsaw puzzle is constructed by independently and uniformly choosing the shape of each ‘jig’ from q possibilities. We are given the shuffled pieces. Then, depending on q, what is the probability that we can reassemble the puzzle uniquely? We say that two solutions of a puzzle are similar if they only differ by a global rotation of the puzzle, permutation of duplicate pieces, and rotation of rotationally symmetric pieces. In this paper, we show that, with high probability, such a puzzle has at least two non-similar solutions when 2 ⩽ q ⩽ 2e−1/2n, all solutions are similar when q ⩾ (2+ϵ)n, and the solution is unique when q = ω(n).

A filling of a closed hyperbolic surface is a set of simple closed geodesics whose complement is a disjoint union of hyperbolic polygons. The systolic length is the length of a shortest essential closed geodesic on the surface. A geodesic is called systolic, if the systolic length is realised by its length. For every $g\geq 2$, we construct closed hyperbolic surfaces of genus $g$ whose systolic geodesics fill the surfaces with complements consisting of only two components. Finally, we remark that one can deform the surfaces obtained to increase the systole.

For an integer q ⩾ 2 and an even integer d, consider the graph obtained from a large complete q-ary tree by connecting with an edge any two vertices at distance exactly d in the tree. This graph has clique number q + 1, and the purpose of this short note is to prove that its chromatic number is Θ((d log q)/log d). It was not known that the chromatic number of this graph grows with d. As a simple corollary of our result, we give a negative answer to a problem of van den Heuvel and Naserasr, asking whether there is a constant C such that for any odd integer d, any planar graph can be coloured with at most C colours such that any pair of vertices at distance exactly d have distinct colours. Finally, we study interval colouring of trees (where vertices at distance at least d and at most cd, for some real c > 1, must be assigned distinct colours), giving a sharp upper bound in the case of bounded degree trees.

By Smith’s theorem, if a cubic graph has a Hamiltonian cycle, then it has a second Hamiltonian cycle. Thomason [‘Hamilton cycles and uniquely edge-colourable graphs’, Ann. Discrete Math.3 (1978), 259–268] gave a simple algorithm to find the second cycle. Thomassen [private communication] observed that if there exists a polynomially bounded algorithm for finding a second Hamiltonian cycle in a cubic cyclically 4-edge connected graph $G$, then there exists a polynomially bounded algorithm for finding a second Hamiltonian cycle in any cubic graph $G$. In this paper we present a class of cyclically 4-edge connected cubic bipartite graphs $G_{i}$ with $16(i+1)$ vertices such that Thomason’s algorithm takes $12(2^{i}-1)+3$ steps to find a second Hamiltonian cycle in $G_{i}$.

We follow the example of Tutte in his construction of the dichromate of a graph (i.e. the Tutte polynomial) as a unification of the chromatic polynomial and the flow polynomial in order to construct a new polynomial invariant of maps (graphs embedded in orientable surfaces). We call this the surface Tutte polynomial. The surface Tutte polynomial of a map contains the Las Vergnas polynomial, the Bollobás–Riordan polynomial and the Krushkal polynomial as specializations. By construction, the surface Tutte polynomial includes among its evaluations the number of local tensions and local flows taking values in any given finite group. Other evaluations include the number of quasi-forests.

A long-standing conjecture of Richter and Thomassen states that the total number of intersection points between any n simple closed Jordan curves in the plane, so that any pair of them intersect and no three curves pass through the same point, is at least (1−o(1))n 2.

We confirm the above conjecture in several important cases, including the case (1) when all curves are convex, and (2) when the family of curves can be partitioned into two equal classes such that each curve from the first class touches every curve from the second class. (Two closed or open curves are said to be touching if they have precisely one point in common and at this point the two curves do not properly cross.)

An important ingredient of our proofs is the following statement. Let S be a family of n open curves in ℝ2, so that each curve is the graph of a continuous real function defined on ℝ, and no three of them pass through the same point. If there are nt pairs of touching curves in S, then the number of crossing points is $\Omega(nt\sqrt{\log t/\log\log t})$ .

We find new properties of the topological transition polynomial of embedded graphs, Q(G). We use these properties to explain the striking similarities between certain evaluations of Bollobás and Riordan's ribbon graph polynomial, R(G), and the topological Penrose polynomial, P(G). The general framework provided by Q(G) also leads to several other combinatorial interpretations these polynomials. In particular, we express P(G), R(G), and the Tutte polynomial, T(G), as sums of chromatic polynomials of graphs derived from G, show that these polynomials count k-valuations of medial graphs, show that R(G) counts edge 3-colourings, and reformulate the Four Colour Theorem in terms of R(G). We conclude with a reduction formula for the transition polynomial of the tensor product of two embedded graphs, showing that it leads to additional relations among these polynomials and to further combinatorial interpretations of P(G) and R(G).

We prove the existence of a function $f :\mathbb{N} \to \mathbb{N}$ such that the vertices of every planar graph with maximum degree Δ can be 3-coloured in such a way that each monochromatic component has at most f(Δ) vertices. This is best possible (the number of colours cannot be reduced and the dependence on the maximum degree cannot be avoided) and answers a question raised by Kleinberg, Motwani, Raghavan and Venkatasubramanian in 1997. Our result extends to graphs of bounded genus.

Let G be a string graph (an intersection graph of continuous arcs in the plane) with m edges. Fox and Pach proved that G has a separator consisting of $O(m^{3/4}\sqrt{\log m})$ vertices, and they conjectured that the bound of $O(\sqrt m)$ actually holds. We obtain separators with $O(\sqrt m \,\log m)$ vertices.

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.

This paper discusses a flaw in Murasugi–Przytycki’s Memoir ‘An index of a graph with applications to knot theory’ [Mem. Amer. Math. Soc. 106 (1993)]. We point out and partly fix a gap occurring in the proof of Murasugi–Przytycki’s braid index inequalities involving the graph index. We explain why their notion of index fails to precisely reflect the reduction of Seifert circles by their diagram move, and redefine the index to account for that discrepancy.

After summarizing from previous papers the definitions of the concepts associated with nets, i.e. triples of 6-transpositions in the Monster up to braiding, we give some results.