Shortest path embeddings of graphs on surfaces

The classical theorem of F\'{a}ry states that every planar graph can be represented by an embedding in which every edge is represented by a straight line segment. We consider generalizations of F\'{a}ry's theorem to surfaces equipped with Riemannian metrics. In this setting, we require that every edge is drawn as a shortest path between its two endpoints and we call an embedding with this property a shortest path embedding. The main question addressed in this paper is whether given a closed surface S, there exists a Riemannian metric for which every topologically embeddable graph admits a shortest path embedding. This question is also motivated by various problems regarding crossing numbers on surfaces. We observe that the round metrics on the sphere and the projective plane have this property. We provide flat metrics on the torus and the Klein bottle which also have this property. Then we show that for the unit square flat metric on the Klein bottle there exists a graph without shortest path embeddings. We show, moreover, that for large g, there exist graphs G embeddable into the orientable surface of genus g, such that with large probability a random hyperbolic metric does not admit a shortest path embedding of G, where the probability measure is proportional to the Weil-Petersson volume on moduli space. Finally, we construct a hyperbolic metric on every orientable surface S of genus g, such that every graph embeddable into S can be embedded so that every edge is a concatenation of at most O(g) shortest paths.Comment: 22 pages, 11 figures: Version 3 is updated after comments of reviewer

Constructions of Large Graphs on Surfaces

We consider the degree/diameter problem for graphs embedded in a surface, namely, given a surface $\Sigma$ and integers $\Delta$ and $k$, determine the maximum order $N(\Delta,k,\Sigma)$ of a graph embeddable in $\Sigma$ with maximum degree $\Delta$ and diameter $k$. We introduce a number of constructions which produce many new largest known planar and toroidal graphs. We record all these graphs in the available tables of largest known graphs. Given a surface $\Sigma$ of Euler genus $g$ and an odd diameter $k$, the current best asymptotic lower bound for $N(\Delta,k,\Sigma)$ is given by $$\sqrt{\frac{3}{8}g}\Delta^{\lfloor k/2\rfloor}.$$ Our constructions produce new graphs of order \begin{cases}6\Delta^{\lfloor k/2\rfloor}& \text{if $\Sigma$ is the Klein bottle}\\ \(\frac{7}{2}+\sqrt{6g+\frac{1}{4}}\)\Delta^{\lfloor k/2\rfloor}& \text{otherwise,}\end{cases} thus improving the former value by a factor of 4.Comment: 15 pages, 7 figure

Grafos com poucos cruzamentos e o número de cruzamentos do Kp,q em superfícies topológicas

Orientador: Orlando LeeTese (doutorado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: O número de cruzamentos de um grafo G em uma superfície ? é o menor número de cruzamentos de arestas dentre todos os possíveis desenhos de G em ?. Esta tese aborda dois problemas distintos envolvendo número de cruzamentos de grafos: caracterização de grafos com número de cruzamentos igual a um e determinação do número de cruzamentos do Kp,q em superfícies topológicas. Para grafos com número de cruzamentos um, apresentamos uma completa caracterização estrutural. Também desenvolvemos um algoritmo "prático" para reconhecer estes grafos. Em relação ao número de cruzamentos do Kp,q em superfícies, mostramos que para um inteiro positivo p e uma superfície ? fixos, existe um conjunto finito D(p,?) de desenhos "bons" de grafos bipartidos completos Kp,r (possivelmente variando o r) tal que, para todo inteiro q e todo desenho D de Kp,q, existe um desenho bom D' de Kp,q obtido através de duplicação de vértices de um desenho D'' em D(p,?) tal que o número de cruzamentos de D' é menor ou igual ao número de cruzamentos de D. Em particular, para todo q suficientemente grande, existe algum desenho do Kp,q com o menor número de cruzamentos possível que é obtido a partir de algum desenho de D(p,?) através da duplicação de vértices do mesmo. Esse resultado é uma extensão de outro obtido por Cristian et. al. para esferaAbstract: The crossing number of a graph G in a surface ? is the least amount of edge crossings among all possible drawings of G in ?. This thesis deals with two problems on crossing number of graphs: characterization of graphs with crossing number one and determining the crossing number of Kp,q in topological surfaces. For graphs with crossing number one, we present a complete structural characterization. We also show a "practical" algorithm for recognition of such graphs. For the crossing number of Kp,q in surfaces, we show that for a fixed positive integer p and a fixed surface ?, there is a finite set D(p,?) of good drawings of complete bipartite graphs Kp,r (with distinct values of r) such that, for every positive integer q and every good drawing D of Kp,q, there is a good drawing D' of Kp,q obtained from a drawing D'' of D(p,?) by duplicating vertices of D'' and such that the crossing number of D' is at most the crossing number of D. In particular, for any large enough q, there exists some drawing of Kp,q with fewest crossings which can be obtained from a drawing of D(p,?) by duplicating vertices. This extends a result of Christian et. al. for the sphereDoutoradoCiência da ComputaçãoDoutor em Ciência da Computação2014/14375-9FAPES