[...]... couple of nontrivial results and give two interesting applications We warn the reader that the terminology in graph theory lacks universality, although this improved a little after the book by Harary [Har69] appeared 1 See [Wil86] and [BigLW76] D Jungnickel, Graphs, Networks and Algorithms, Algorithms and Computation in Mathematics 5, DOI 10.1007/97 8-3 -6 4 2-3 227 8-5 1, © Springer-Verlag Berlin Heidelberg... K¨nigsberg bridge problem o 1.1 Graphs, Subgraphs and Factors A graph G is a pair G = (V, E) consisting of a finite2 set V = ∅ and a set E of two-element subsets of V The elements of V are called vertices An element e = {a, b} of E is called an edge with end vertices a and b We say that a and b are incident with e and that a and b are adjacent or neighbors of each e other, and write e = ab or a — b Let... universities I thank my students and assistants and the students who attended the workshop mentioned above for their constant attention and steady interest Thanks are due, in particular, to Priv.-Doz Dr Dirk Hachenberger and Prof Dr Alexander Pott who read the entire manuscript of the (German) third edition with critical accuracy; the remaining errors are my responsibility Augsburg Dieter Jungnickel Contents When... 1 2 5 13 15 21 25 29 2 Algorithms and Complexity 2.1 Algorithms 2.2 Representing Graphs 2.3 The Algorithm of Hierholzer 2.4 How to Write Down Algorithms 2.5 The Complexity of Algorithms 2.6 Directed Acyclic Graphs 2.7 An Introduction to NP-completeness 2.8 Five NP-complete Problems ... [CamLi91] and [BetJL99] We will limit our look at SRG’s in this book to a few exercises Exercise 1.1.3 Draw the graphs Tn for n = 3, 4, 5 and show that Tn has parameters a = 2n − 4, c = n − 2 and d = 4, where a is the degree of any 1.2 Paths, Cycles, Connectedness, Trees 5 vertex, c is the number of vertices adjacent to both x and y if x and y are adjacent, and d is the number of vertices adjacent to x and. .. 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 (We leave it to the reader to show that [G] is uniquely determined.) Then we have the following theorem due to Bondy and Chv´tal... Dissections: Dilworth’s Theorem 7.6 Parallelisms: Baranyai’s Theorem 7.7 Supply and Demand: The Gale-Ryser Theorem 219 219 224 228 234 239 243 246 8 Connectivity and Depth First Search 8.1 k-connected Graphs 8.2 Depth First Search 8.3 2-connected Graphs 8.4 Depth First Search for Digraphs 8.5 Strongly Connected... 1.2.12 suggests) and get a tree G on V It is easy to verify that v = v(G) and thus πV (G) = w To prove injectivity, let G and H be two trees on {1, , n} and suppose πV (G) = πV (H) Now let v be the smallest element of V which does not occur in πV (G) Then Lemma 1.2.12 implies that v = v(G) = v(H) Thus G and H both contain the edge e = vw, where w is the first entry of πV (G) Then G and H are both... right hand side is even, the number of odd terms deg v in the sum on the left hand side must also be even 4 1 Basic Graph Theory Fig 1.4 A factorization of K6 If all vertices of a graph G have the same degree (say r), G is called a regular graph, more precisely an r-regular graph The graph Kn is (n − 1)regular, the graph Km,n is regular only if m = n (in which case it is n-regular) A k-factor is a k-regular... vertices (that is, |V | = n) and all two-element subsets of V as edges The complete bipartite graph Km,n has as vertex set the disjoint union of a set V1 with m elements and a set V2 with n elements; edges are all the sets {a, b} with a ∈ V1 and b ∈ V2 We will often illustrate graphs by pictures in the plane The vertices of a graph G = (V, E) are represented by (bold type) points and the edges by lines . well, a sec- ond edition became necessary in 1990. This second edition was only slightly changed (there were only a few corrections and some additions made, includ- ing a further appendix and a number. appeared. 1 See [Wil86 ]and[ BigLW76]. D. Jungnickel, Graphs, Networks and Algorithms, Algorithms and Computation in Mathematics 5, DOI 10.1007/97 8-3 -6 4 2-3 227 8-5 1, © Springer-Verlag Berlin Heidelberg 2013 1 2. called a walk;ifv 0 = v n , one speaks of a closed walk.Awalkforwhichthee i are distinct is called a trail,andaclosedwalk with distinct edges is a closed trail. If, in addition, the v j are distinct,