Abstract
We begin this chapter by considering the following three combinatorial problems.
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- 1.
Actually, as we shall see later, Euler is usually credited as being the founder of Graph Theory.
- 2.
After Václav J. Havel , Czech mathematician, and Seifollah L. Hakimi , Iranian-American mathematician, both of the twentieth century.
- 3.
After Julius P. C. Petersen, Danish mathematician of the nineteenth century.
- 4.
For another approach to this problem, see Problem 5, page 49.
- 5.
After Dénes König , Hungarian mathematician of the twentieth century.
- 6.
Note that, in principle, this definition of path does not coincide with the one given in the text. Nevertheless, we have chosen to translate it according to the original.
- 7.
We call the reader’s attention for not confusing notations w(G) for the weight of a weighted graph and ω(G) for the clique number of a graph; the first uses the Latin letter w, whereas the second uses the Greek letter ω ( omega).
- 8.
In terms of Logic, an implication is true whenever it premise is false. See Chapter 1 of [34], for instance.
- 9.
Note that this is precisely Turán’s number T(n; l) when n = (l − 1)q.
References
T. Apostol, Calculus, Vol. 2 (Wiley, New York, 1967)
R. Diestel, Graph Theory (Springer, New York, 2000)
S.B. Feitosa, Turán’s Theorem (in Portuguese) (Classnotes, 2006)
N. Hartsfield, G. Ringel, Pearls in Graph Theory (Academic Press, San Diego, 1990)
E. Scheinerman, Mathematics: A Discrete Introduction, 3rd edn. (Brooks Cole, Boston, 2012)
P. Turán, An extremal problem in graph theory. Mat. Fiz. Lapok 41, 435–452 (1941)
J.H. Van Lint, R.M. Wilson, Combinatorics (Cambridge University Press, Cambridge, 2001)
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Caminha Muniz Neto, A. (2018). A Glimpse on Graph Theory. In: An Excursion through Elementary Mathematics, Volume III. Problem Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-77977-5_5
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