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Basic Graph Theory

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Graphs, Networks and Algorithms

Part of the book series: Algorithms and Computation in Mathematics ((AACIM,volume 5))

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Abstract

Graph theory began in 1736 when Leonhard Euler solved the well-known Königsberg bridge problem. This problem asked for a circular walk through the town of Königsberg (now Kaliningrad) in such a way as to cross over each of the seven bridges spanning the river Pregel once, and only once. Euler realized that the precise shapes of the island and the other three territories involved are not important; the solvability depends only on their connection properties. This led to the abstract notion of a graph. Actually, Euler proved much more: he gave a necessary and sufficient condition for an arbitrary graph to admit such a circular walk. His theorem is one of the highlights in the introductory Chap. 1, where we deal with some of the most fundamental notions in graph theory: paths, cycles, connectedness, 1-factors, trees, Euler tours and Hamiltonian cycles, the travelling salesman problem, drawing graphs in the plane, and directed graphs. We will also see a first application, namely setting up a schedule for a tournament, say in soccer or basketball, where each of the 2n participating teams should play against each of the other teams exactly twice, once at home and once away.

It is time to get back to basics.

John Major

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Notes

  1. 1.

    See [Wil86] and [BigLW76].

  2. 2.

    In graph theory, infinite graphs are studied as well. However, we restrict ourselves in this book—like [Har69]—to the finite case.

  3. 3.

    Some authors denote the structure we call a multigraph by graph; graphs according to our definition are then called simple graphs. Moreover, sometimes even edges e for which J(e) is a set {a} having only one element are admitted; such edges are then called loops. The corresponding generalization of multigraphs is often called a pseudograph.

  4. 4.

    Sometimes one also uses the term Eulerian cycle, even though an Euler tour usually contains vertices more than once.

  5. 5.

    It seems that the first known knight’s tours go back more than a thousand years to the Islamic and Indian world around 840–900. The first examples in the modern European literature occur in 1725 in Ozanam’s book [Oza25], and the first mathematical analysis of knight’s tours appears in a paper presented by Euler to the Academy of Sciences at Berlin in 1759 [Eul66]. See the excellent website by Jelliss [Jel03]; and [Wil89], an interesting account of the history of Hamiltonian graphs.

  6. 6.

    The distinction between easy and hard problems can be made quite precise; we will explain this in Chap. 2.

  7. 7.

    In the definition of planar graphs, one often allows not only line segments, but curves as well. However, this does not change the definition of planarity as given above, see [Wag36]. For multigraphs, it is necessary to allow curves.

  8. 8.

    Note that we introduce only one edge wx, even if x was adjacent to both u and v, which is the appropriate operation in our context. However, there are occasions where it is actually necessary to introduce two parallel edges wx instead, so that a contracted graph will in general become a multigraph.

  9. 9.

    This section will not be used in the remainder of the book and may be skipped during the first reading.

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  1. Institut für Mathematik, Universität Augsburg, Augsburg, Germany

    Dieter Jungnickel

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Jungnickel, D. (2013). Basic Graph Theory. In: Graphs, Networks and Algorithms. Algorithms and Computation in Mathematics, vol 5. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32278-5_1

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