All the shapes of spaces: a census of small 3-manifolds

In this work we present a complete (no misses, no duplicates) census for closed, connected, orientable and prime 3-manifolds induced by plane graphs with a bipartition of its edge set (blinks) up to $k=9$ edges. Blinks form a universal encoding for such manifolds. In fact, each such a manifold is a subtle class of blinks, \cite{lins2013B}. Blinks are in 1-1 correpondence with {\em blackboard framed links}, \cite {kauffman1991knots, kauffman1994tlr} We hope that this census becomes as useful for the study of concrete examples of 3-manifolds as the tables of knots are in the study of knots and links.Comment: 31 pages, 17 figures, 38 references. In this version we introduce some new material concerning composite manifold

Smallest snarks with oddness 4 and cyclic connectivity 4 have order 44

The family of snarks -- connected bridgeless cubic graphs that cannot be 3-edge-coloured -- is well-known as a potential source of counterexamples to several important and long-standing conjectures in graph theory. These include the cycle double cover conjecture, Tutte's 5-flow conjecture, Fulkerson's conjecture, and several others. One way of approaching these conjectures is through the study of structural properties of snarks and construction of small examples with given properties. In this paper we deal with the problem of determining the smallest order of a nontrivial snark (that is, one which is cyclically 4-edge-connected and has girth at least 5) of oddness at least 4. Using a combination of structural analysis with extensive computations we prove that the smallest order of a snark with oddness at least 4 and cyclic connectivity 4 is 44. Formerly it was known that such a snark must have at least 38 vertices [J. Combin. Theory Ser. B 103 (2013), 468--488] and one such snark on 44 vertices was constructed by Lukot'ka et al. [Electron. J. Combin. 22 (2015), #P1.51]. The proof requires determining all cyclically 4-edge-connected snarks on 36 vertices, which extends the previously compiled list of all such snarks up to 34 vertices [J. Combin. Theory Ser. B, loc. cit.]. As a by-product, we use this new list to test the validity of several conjectures where snarks can be smallest counterexamples.Comment: 21 page

Decomposition of Geometric Set Systems and Graphs

We study two decomposition problems in combinatorial geometry. The first part deals with the decomposition of multiple coverings of the plane. We say that a planar set is cover-decomposable if there is a constant m such that any m-fold covering of the plane with its translates is decomposable into two disjoint coverings of the whole plane. Pach conjectured that every convex set is cover-decomposable. We verify his conjecture for polygons. Moreover, if m is large enough, we prove that any m-fold covering can even be decomposed into k coverings. Then we show that the situation is exactly the opposite in 3 dimensions, for any polyhedron and any $m$ we construct an m-fold covering of the space that is not decomposable. We also give constructions that show that concave polygons are usually not cover-decomposable. We start the first part with a detailed survey of all results on the cover-decomposability of polygons. The second part investigates another geometric partition problem, related to planar representation of graphs. The slope number of a graph G is the smallest number s with the property that G has a straight-line drawing with edges of at most s distinct slopes and with no bends. We examine the slope number of bounded degree graphs. Our main results are that if the maximum degree is at least 5, then the slope number tends to infinity as the number of vertices grows but every graph with maximum degree at most 3 can be embedded with only five slopes. We also prove that such an embedding exists for the related notion called slope parameter. Finally, we study the planar slope number, defined only for planar graphs as the smallest number s with the property that the graph has a straight-line drawing in the plane without any crossings such that the edges are segments of only s distinct slopes. We show that the planar slope number of planar graphs with bounded degree is bounded.Comment: This is my PhD thesi

The Four Color Problem: The Journey to a Proof and the Results of the Study

The four color problem was one of the most difficult to prove problems for 150 years. It took several failed proofs and advancement in technology and techniques for the final proof to become possible. Some notable men include De Morgan first writing about the problem, Kempe giving the first proof, Heawood showing the flaws in Kempe’s work as well as making advancements of his own. The first actual proof of the problem is then discussed, as well as it’s shortcomings and the work done by other mathematicians to show improvements on them. The total of this work has lead to numerous great leaps in mathematics including the creation of the branch known as graph theory. This one problem also revolutionized proof writing, being the first to use a computer as an essential part of the proving process

Graph Theory

This was a workshop on graph theory, with a comprehensive approach. Highlights included the emerging theories of sparse graph limits and of infinite matroids, new techniques for colouring graphs on surfaces, and extensions of graph minor theory to directed graphs and to immersion

Desenvolvimentos da Conjetura de Fulkerson

Orientador: Christiane Neme CamposDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: Em 1971, Fulkerson propôs a seguinte conjetura: todo grafo cúbico sem arestas de corte admite seis emparelhamentos perfeitos tais que cada aresta do grafo pertence a exatamente dois destes emparelhamentos. A Conjetura de Fulkerson tem desafiado pesquisadores desde sua publicação. Esta conjetura é facilmente verificada para grafos cúbicos 3-aresta-coloráveis. Portanto, a dificuldade do problema reside em estabelecer a conjetura para grafos cúbicos sem arestas de corte que não possuem 3-coloração de arestas. Estes grafos são chamados snarks. Nesta dissertação, a Conjetura de Fulkerson e os snarks são introduzidos com ¿ênfase em sua história e resultados mais relevantes. Alguns resultados relacionados à Conjetura de Fulkerson são apresentados, enfatizando suas conexões com outras conjeturas. Um breve histórico do Problema das Quatro Cores e suas relações com snarks também são apresentados. Na segunda parte deste trabalho, a Conjetura de Fulkerson é verificada para algumas famílias infinitas de snarks construídas com o método de Loupekine, utilizando subgrafos do Grafo de Petersen. Primeiramente, mostramos que a família dos LP0-snarks satisfaz a Conjetura de Fulkerson. Em seguida, generalizamos este resultado para a família mais abrangente dos LP1-snarks. Além disto, estendemos estes resultados para Snarks de Loupekine construídos com subgrafos de snarks diferentes do Grafo de PetersenAbstract: In 1971, Fulkerson proposed a conjecture that states that every bridgeless cubic graph has six perfect matchings such that each edge of the graph belongs to precisely two of these matchings. Fulkerson's Conjecture has been challenging researchers since its publication. It is easily verified for 3-edge-colourable cubic graphs. Therefore, the difficult task is to settle the conjecture for non-3-edge-colourable bridgeless cubic graphs, called snarks. In this dissertation, Fulkerson's Conjecture and snarks are presented with emphasis in their history and remarkable results. We selected some results related to Fulkerson's Conjecture, emphasizing their reach and connections with other conjectures. It is also presented a brief history of the Four-Colour Problem and its connections with snarks. In the second part of this work, we verify Fulkerson's Conjecture for some infinite families of snarks constructed with Loupekine's method using subgraphs of the Petersen Graph. More specifically, we first show that the family of LP0-snarks satisfies Fulkerson's Conjecture. Then, we generalise this result by proving that Fulkerson's Conjecture holds for the broader family of LP1-snarks. We also extend these results to even more general Loupekine Snarks constructed with subgraphs of snarks other than the Petersen GraphMestradoCiência da ComputaçãoMestre em Ciência da Computaçã

Orientations des graphes (structures et algorithmes)

Orienter un graphe c'est remplacer chaque arête par un arc de mêmes extrémités. On s'intéresse à la connexité du graphe orienté ainsi obtenu. L'orientation avec des contraintes d'arc-connexité est maintenant comprise en profondeur mais très peu de résultats sont connus en terme de sommet-connexité. La conjecture de Thomassen avance que les graphes suffisament sommet-connexes ont une orientation k-sommet-connexe. De plus, la conjecture de Frank propose une caractérisation des graphes qui admettent une telle orientation. Les résultats de cette thèse s'articulent autour des notions d'orientation, de packing, de connexité et de matroïde. D'abord, nous infirmons une conjecture de Recski sur la décomposition d'un graphe en arbres ayant des orientations avec degrés entrants prescrits. Nous prouvons également un nouveau résultat sur le packing d'arborescences enracinées avec contraintes de matroïdes. Ceci généralise un résultat fondamental d'Edmonds. Enfin, nous démontrons un nouveau théorème de packing sur les bases des matroïdes de dénombrement qui nous permet d'améliorez le seul résultat connu sur la conjecture de Thomassen. D'autre part, nous donnons une construction et un théorème d'augmentation pour une famille de graphes liée à la conjecture de Frank. En conclusion, nous réfutons la conjecture de Frank et prouvons que, pour tout entier k >= 3, décider si un graphe a une orientation k-sommet-connexe est un problème NP-complet.Orienting an undirected graph means replacing each edge by an arc with the same ends. We investigate the connectivity of the resulting directed graph. Orientations with arc-connectivity constraints are now deeply understood but very few results are known in terms of vertex-connectivity. Thomassen conjectured that sufficiently highly vertex-connected graphs have a k-vertex- connected orientation while Frank conjectured a characterization of the graphs admitting such an orientation. The results of this thesis are structures around the concepts of orientation, packing, connectivity and matroid. First, we disprove a conjecture of Recski on decomposing a graph into trees having orientations with specified indegrees. We also prove a new result on packing rooted arborescences with matroid constraints. This generalizes a fundamental result of Edmonds. Moreover, we show a new packing theorem for the bases of count matroids that induces an improvement of the only known result on Thomassen's conjecture. Secondly, we give a construction and an augmentation theorem for a family of graphs related to Frank's conjecture. To conclude, we disprove the conjecture of Frank and prove that, for every integer k >= 3, the problem of deciding whether a graph admits a k-vertex-orientation is NP-complete.SAVOIE-SCD - Bib.électronique (730659901) / SudocGRENOBLE1/INP-Bib.électronique (384210012) / SudocGRENOBLE2/3-Bib.électronique (384219901) / SudocSudocFranceF

Discrete Geometry

A number of important recent developments in various branches of discrete geometry were presented at the workshop. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as topology, combinatorics or algebraic geometry. The open questions abound and many of the results presented were obtained by young researchers, confirming the great vitality of discrete geometry