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Project Synopsis: Topological Graph Theory and - Heawood Problem

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Project Synopsis: Topological Graph Theory and - Heawood Problem Project Synopsis: Topological Graph theory and - Heawood Problem Sreekanth D (13142) Background Topological graph theory is a branch of graph theory that studies graphs as topological spaces, their embeddings on surfaces and other properties alongside the combinatorial and algebraic definition. The primary objective of topological graph theory is to study graph embeddings on surfaces, which in layman's terms, pertains to understanding whether a given graph can be drawn on a surface without crossings. An embedding of the complete bipartite graph K3;3 on a torus. The breakthrough of this field was the Ringel-Young solution (1968) [1] of Heawood problem, a classic map-colouring problem. In early 1987, the frontiers of topological graph theory began advancing in numerous different directions. Some of the frontier topics include algorithms for imbedding problems, covering-space constructions, enumerative analysis of imbedding distributions, genus of groups, representation of higher-dimensional manifolds by graphs, and VLSI layouts. Synopsis The primary focus of this project will be on graph embeddings [1] on surfaces. First, we will study planar embeddings and the Kuratowski's theorem [1]. Kuratowski's theorem states that a graph is planar if and only if it contains no embedding of the complete bipartite graph on six vertices or the complete graph on five vertices. After that, we will study embedding of graphs on other surfaces (orientable or non-orientable) along with a quick glance on some relevant topics like voltage graph [1], which is a combinatorial form of covering spaces [2] (of graphs). Finally, we will review the Heawood conjecture [1] and a representable sample of cases whose solutions are most readily generalized to other embedding problems. And if time permits, we will include an in-depth analysis of some important algorithms [1, 3] on graph embeddings and genus calculation. Heawood Problem The chromatic number chr(G) is defined to be the smallest number n such that G has an n-colouring. The chromatic number chr(S) of a surface S is equal to the maximum of the set of chromatic numbers of the simplicial graphs that can be embedded in S. The determination of the chromatic number of the surfaces other than the sphere is called the Heawood problem. It was proven that for all surfaces S except Klein bottle, chr(S) = H(S), the Heawood number of S (defined in (1)). When S is the sphere, then chr(S) = 4 = H(S), which is the gist of the famous four-colouring theorem whose discussion is avoided in this project due to its predominance in combinatorics than topology. The original solution to the Heawood problem occupies about 300 journal pages, spread over numerous separate articles, and it requires several different kinds of current graphs, whose properties are individually derived. Due to these constraints, we will narrow down our exploration of the Heawood problem to a basic overview on the implementation of the Ringel-Young solution. The initial step in this path is to derive the Heawood inequality (below). Theorem. Let S be a closed surface with Euler characteristic c ≤ 1. Then p 7 + 49 − 24c chr(S) ≤ (1) 2 The expression on the right is called the Heawood number H(S) of S. After that, we will use Heawood inequality to reduce the Heawood problem to finding the genus (g) and the crosscap number (g) of every complete graph and then try to solve it. References [1] J. L. Gross and T. W. Tucker, Topological graph theory. Dover Publications, Inc., Mineola, NY, 2001. Reprint of the 1987 original [Wiley, New York; MR0898434 (88h:05034)] with a new preface and supplementary bibliography. [2] J. R. Munkres, Topology: a first course. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1975. [3] T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms, 3rd Edition (MIT Press). The MIT Press, 3rd ed., 7 2009..
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