Graphs, Networks and Algorithms by Dieter Jungnickel (auth.)

By Dieter Jungnickel (auth.)

From the stories of the former variants

".... The ebook is a firstclass textbook and appears to be like necessary for everyone who has to coach combinatorial optimization. it's very priceless for college students, lecturers, and researchers during this region. the writer reveals a awesome synthesis of great and fascinating mathematical effects and functional purposes. ... the writer can pay a lot realization to the inclusion of well-chosen routines. The reader doesn't stay helpless; strategies or not less than tricks are given within the appendix. apart from a few small simple mathematical and algorithmic wisdom the e-book is self-contained. ..." K.Engel, Mathematical stories 2002

The enormous improvement attempt of this article, regarding a number of variants and trailing within the context of varied workshops, collage classes and seminar sequence, in actual fact exhibits via during this re-creation with its transparent writing, solid enterprise, finished insurance of crucial conception, and well-chosen functions. The proofs of significant effects and the illustration of key algorithms in a Pascal-like notation let this e-book for use in a high-level undergraduate or low-level graduate path on graph thought, combinatorial optimization or laptop technology algorithms. The well-worked suggestions to workouts are a true bonus for self research through scholars. The booklet is extremely advised. P .B. Gibbons, Zentralblatt für Mathematik 2005

Once back, the hot version has been completely revised. specifically, a few extra fabric has been further: extra on NP-completeness (especially on dominating sets), a bit at the Gallai-Edmonds constitution idea for matchings, and a few dozen extra workouts – as constantly, with strategies. additionally, the part at the 1-factor theorem has been thoroughly rewritten: it now provides a brief direct evidence for the extra basic Berge-Tutte formulation. numerous fresh examine advancements are mentioned and numerous references were added.

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If we identify two adjacent vertices u and v in a graph G, we get an elementary contraction of G; more precisely, we omit u and v and replace them by a new vertex w which is adjacent to all vertices which were adjacent to u or v before;8 the resulting graph is usually denoted by G/e, where e = uv. 11 also shows a contraction of K3,3 . A graph G is called contractible to a graph H if H arises from G by a sequence of elementary contractions. For the proof of the following theorem see [Wag37, Aig84], or [HarTu65].

3 Let G be a graph with n ≥ 3 vertices. If each vertex of G has degree at least n/2, then G is Hamiltonian. 4 Hamiltonian Cycles 17 Fig. 1 to derive further sufficient conditions for the existence of a Hamiltonian cycle; in particular, they obtained the earlier result of Las Vergnas [Las72] in this way. We also refer the reader to [Har69, Ber73, Ber78, GonMi84, Chv85] for more results about Hamiltonian graphs. 4 Let G be a graph with n vertices and m edges, and assume m ≥ 12 (n − 1)(n − 2) + 2. 2 to show that G is Hamiltonian.

16 1 Basic Graph Theory We first prove a theorem from which we can derive several sufficient conditions on the sequence of degrees in a graph. Let G be a graph on n vertices. If G contains non-adjacent vertices u and v such that deg u + deg v ≥ n, we add the edge uv to G. We continue this procedure until we get a graph [G], in which, for any two non-adjacent vertices x and y, we always have deg x + deg y < n. The graph [G] is called the closure of G. ) Then we have the following theorem due to Bondy and Chv´atal [BonCh76].

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