Cambridge University Press, 2009. — 368 p. — (Encyclopedia of Mathematics and its Applications 128) — ISBN 978-0-521-80230-7.The origins of topological graph theory lie in the 19th century, largely with the four colour problem and its extension to higher-order surfaces – the Heawood map problem. With the explosive growth of topology in the early 20th century, mathematicians like Veblen, Rado and Papakyriakopoulos provided foundational results for understanding surfaces combinatorially and algebraically. Kuratowski, MacLane and Whitney in the 1930s approached the four colour problem as a question about the structure of graphs that can be drawn without edge-crossings in the plane. Kuratowski’s theorem characterizing planarity by two obstructions is the most famous, and its generalization to the higher-order surfaces became an influential unsolved problem. Contents -Introduction -Embedding graphs on surfaces -Maximum genus -Distribution of embeddings -Algorithms and obstructions for embeddings -Colouring graphs on surfaces -Crossing numbers -Representing graphs and maps -Enumerating coverings -Symmetric maps -The genus of a group -Embeddings and geometries -Embeddings and designs -Infinite graphs and planar maps -Open problems
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