Combinatorial Problems and Exercises
The main purpose of this book is to provide help in learning existing techniques in combinatorics. The most effective way of learning such techniques is to solve exercises and problems. This book presents all the material in the form of problems and series of problems (apart from some general comments at the beginning of each chapter). In the second part, a hint is given for each exercise, which contains the main idea necessary for the solution, but allows the reader to practice theechniques by completing the proof. In the third part, a full solution is provided for each problem. This book will be useful to those students who intend to start research in graph theory, combinatorics or their applications, and for those researchers who feel that combinatorial techniques mightelp them with their work in other branches of mathematics, computer science, management science, electrical engineering and so on. For background, only the elements of linear algebra, group theory, probability and calculus are needed.
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1-factor 2-connected adjacent arises assertion assume automorphism belongs bipartite circuit claim classes clearly color complete components connected Consider consists construction contains contradiction Conversely corresponding counted cover cycle defined degree denote Determine digraph directed disjoint distinct edges edges of G eigenvalues elements endpoints equal equivalent exactly exists faces fact fc-coloration Figure fixed follows formula give given graph G hand Hence holds hypergraph identity implies induction joined least length Let G matching Math maximum means meet minimal Moreover neighbors Observe obtain obviously occur orientation otherwise pair partition path permutation points possible problem Prove random remove respectively resulting root satisfies sequence side Similarly simple graph solution spanning starting steps subgraph subset Suppose theorem theory tree triangle trivial unique vertices walk
Page 45 - A graph is planar if and only if it contains no subdivision of...
Page 31 - Along a speed track there are some gas-stations. The total amount of gasoline available in them is equal to what our car (which has a very large tank) needs for going around the track. Prove that there is a gas-station such that if we start there with an empty tank, we shall be able to go around the track without running out of gasoline.
Page 11 - Those techniques whose absence has been disapproved of above await their discoverers. So underdevelopment is not a case against, but rather for, directing young scientists toward a given field.
Page 9 - I could not resist, however, to working out a series of exercises on random walks on graphs, and their relations to eigenvalues, expansion properties, and electrical resistance (this area has classical roots but has grown explosively in the last few years).
Page 11 - As long as the main questions have not been formulated and the abstractions to a general level have not been carried through, there is no way to distinguish between interesting and less interesting results — except on an aesthetic basis, which is, of course, too subjective.
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