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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2-coloration 2-connected 3-regular a-critical adjacent automorphism group bipartite graph classes color combinatorial complete graph components of G connected graph Consider contradiction cycle defined degree at least denote the number digraph disjoint edges of G eigenvalues eigenvector elements endpoints exactly fc-coloration follows formula G contains G is connected graph G Hamiltonian circuit Hamiltonian path Hence hint hypergraph implies independent set induced subgraph induction hypothesis integers isomorphic joined Let G Math matrix maximum independent set maximum matching Menger's theorem minimal monochromatic neighbors number of edges number of partitions number of points obviously odd circuit orientation pair permutation planar planar graph points of degree points of G polynomial proves the assertion random walk recurrence relation remove resulting graph sequence Similarly simple graph solution spanning tree strongly connected subgraph of G subset Suppose indirectly Suppose that G theorem triangle trivial vertex vertices whence
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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