Abstract
Graph Theory is an area of modern mathematics with many applications in today’s world, but its roots lie in several recreational puzzles going back to the mid-eighteenth century. This chapter will introduce a few main topics in Graph Theory, drawing upon this history. The first two sections look at ways one can traverse a graph (Eulerian trails and Hamiltonian paths), while the last two sections deal with planar graphs (ones that can be drawn so their edges don’t cross) and graph coloring (graphs whose adjacent vertices have different colors). While connected to a couple of earlier topics, this concluding chapter has a more geometric character, balancing out the algebraic emphasis of the rest of the text.
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Notes
- 1.
See The Truth about Königsberg by Brian Hopkins and Robin J. Wilson in the May 2004 issue, of The College Mathematics Journal for a nice discussion of Euler’s argument. Euler’s paper (and lots more) is in Graph Theory: 1736–1936 (Oxford University Press, 1976) by Norman L. Biggs, E. Keith Lloyd, and Robin J. Wilson.
- 2.
See Joseph Malkevitch’s two 2005 AMS Feature Columns on Euler’s Polyhedral Formula at http://www.ams.org/samplings/feature-column/fcarc-eulers-formula
- 3.
Robin Wilson’s Four Colors Suffice (Princeton University Press, 2013) gives a fascinating and very readable account of the entire history of the four-color problem.
- 4.
This is Alexander Soifer’s minimal counterexample. See The Mathematical Coloring Book (Springer, 2009), p. 182. The Fritsch Graph provides another counterexample on nine vertices.
- 5.
For more information on Gardner’s map and its four-coloring, see the Wolfram MathWorld posting on the Four-Color Theorem at http://mathworld.wolfram.com/Four-ColorTheorem.html.
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Jongsma, C. (2019). Topics in Graph Theory. In: Introduction to Discrete Mathematics via Logic and Proof. Undergraduate Texts in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-25358-5_8
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