Planar graphs : a historical perspective.

The field of graph theory has been indubitably influenced by the study of planar graphs. This thesis, consisting of five chapters, is a historical account of the origins and development of concepts pertaining to planar graphs and their applications. The first chapter serves as an introduction to the history of graph theory, including early studies of graph theory tools such as paths, circuits, and trees. The second chapter pertains to the relationship between polyhedra and planar graphs, specifically the result of Euler concerning the number of vertices, edges, and faces of a polyhedron. Counterexamples and generalizations of Euler\u27s formula are also discussed. Chapter III describes the background in recreational mathematics of the graphs of K5 and K3,3 and their importance to the first characterization of planar graphs by Kuratowski. Further characterizations of planar graphs by Whitney, Wagner, and MacLane are also addressed. The focus of Chapter IV is the history and eventual proof of the four-color theorem, although it also includes a discussion of generalizations involving coloring maps on surfaces of higher genus. The final chapter gives a number of measurements of a graph\u27s closeness to planarity, including the concepts of crossing number, thickness, splitting number, and coarseness. The chapter conclused with a discussion of two other coloring problems - Heawood\u27s empire problem and Ringel\u27s earth-moon problem

Coloring hypergraphs with excluded minors

Hadwiger's conjecture, among the most famous open problems in graph theory, states that every graph that does not contain $K_t$ as a minor is properly $(t-1)$-colorable. The purpose of this work is to demonstrate that a natural extension of Hadwiger's problem to hypergraph coloring exists, and to derive some first partial results and applications. Generalizing ordinary graph minors to hypergraphs, we say that a hypergraph $H_1$ is a minor of a hypergraph $H_2$, if a hypergraph isomorphic to $H_1$ can be obtained from $H_2$ via a finite sequence of vertex- and hyperedge-deletions, and hyperedge contractions. We first show that a weak extension of Hadwiger's conjecture to hypergraphs holds true: For every $t \ge 1$, there exists a finite (smallest) integer $h(t)$ such that every hypergraph with no $K_t$-minor is $h(t)$-colorable, and we prove $$\left\lceil\frac{3}{2}(t-1)\right\rceil \le h(t) \le 2g(t)$$ where $g(t)$ denotes the maximum chromatic number of graphs with no $K_t$-minor. Using the recent result by Delcourt and Postle that $g(t)=O(t \log \log t)$, this yields $h(t)=O(t \log \log t)$. We further conjecture that $h(t)=\left\lceil\frac{3}{2}(t-1)\right\rceil$, i.e., that every hypergraph with no $K_t$-minor is $\left\lceil\frac{3}{2}(t-1)\right\rceil$-colorable for all $t \ge 1$, and prove this conjecture for all hypergraphs with independence number at most $2$. By considering special classes of hypergraphs, the above additionally has some interesting applications for ordinary graph coloring, such as: -graphs of chromatic number $C k t \log \log t$ contain $K_t$-minors with $k$-edge-connected branch-sets, -graphs of chromatic number $C q t \log \log t$ contain $K_t$-minors with modulo-$q$-connected branch sets, -by considering cycle hypergraphs of digraphs we recover known results on strong minors in digraphs of large dichromatic number as special cases.Comment: 15 pages, corrected proof of Proposition

Spin(9)-invariant valuations

The first aim of this thesis is to give a Hadwiger-type theorem for the exceptional Lie group Spin(9). The space of Spin(9)-invariant k-homogeneous valuations is studied through the construction of an exact sequence involving some spaces of differential forms. We present then a description of the spin representation using the properties of the 8-dimensional division algebra of the octonions. Using this description as well as representation-theoretic formulas, we can compute the dimensions of the spaces of differential forms appearing in the exact sequence. Hence we obtain the dimensions of the spaces of k-homogeneous Spin(9)-invariant valuations for k=0,1,...,16. In the second part of this work, we construct one new element for a basis of one of these spaces. It is clear, that the k-th intrinsic volume is also Spin(9)-invariant. The last chapter of this work presents the construction of a new 2-homogeneous Spin(9)-invariant valuation. On a Riemannian manifold (M,g), we construct a valuation by integrating the curvature tensor over the disc bundle. We associate to this valuation on M a family of valuations on the tangent spaces. We show that these valuations are even and homogeneous of degree 2. Moreover, since the valuation on M is invariant under the action of the isometry group of M, the induced valuation on the tangent space in a point p in M is invariant under the action of the stabilisator of p for all p in M. In the special case where M is the octonionic projective plane, this construction yields an even, homogeneous of degree 2, Spin(9)-invariant valuation, whose Klain function is not constant, i.e. which is linearly independent of the second intrinsic volume