Some problems in topological graph theory

Abstract

A list of 31 problems presented here reflects some of the main trends in topological graph theory. INTRODUCTIONDuring the past 10 years or so, about 100 different authors representing a total of about 20 different countries have used recognizably topological methods to obtain graph theoretic results. Although the majority of these authors are best known as graph theorists, many others among them are primarily topologists, algebraists, or computer scientists. The great emphasis on genus in the present collection of problems is about proportional to that in the general literature. Some of the problems are included for their seeming strategic value for obtaining additional results, others because they would strengthen the ties to areas such as group theory or computational complexity, and others mainly due to the longstanding interest of the present authors in their solution.For further information on the context of some of these problems, it may be useful to consult the recent surveys of topological graph theory by Stahl [37] and by White and Beineke [44]. All unexplained graph theoretic terminology here follows the usage of Harary [22].

Group actions on topological graphs

“…If E and G are discrete and G acts freely on E, then by Theorem 2.2.2 in [GT01], there is a cocycle c : (…”

Decentralized Coordination with Local Interactions: Some New Directions

“…Note that this is different from the concept of planar graphs [15] or imbedding graphs in R 2 , where edges are not necessarily mapped to straight lines.…”

On the parameterized complexity of monotone and antimonotone weighted circuit satisfiability

“…We say a (multi)hypergraph H is embeddable in a surface if the bipartite incidence graph obtained from H by replacing each of its hyperedges by a vertex adjacent to all the vertices in the hyperedge is embeddable in that surface. In particular, this definition allows us to speak of (multi)hypergraph of genus g. We refer the reader to [17] for more information on the genus of a graph. The following lemmas will be useful:…”