Combinatorial Problems and Exercises

Front Cover
American Mathematical Soc., 1979 - Mathematics - 639 pages
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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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Contents

III
15
IV
22
V
29
VI
33
VII
40
VIII
45
IX
55
X
62
XXIX
147
XXX
148
XXXI
153
XXXII
156
XXXIII
161
XXXIV
200
XXXV
225
XXXVI
246

XI
65
XII
72
XIII
78
XIV
87
XV
90
XVI
98
XVII
103
XVIII
109
XIX
113
XX
116
XXI
118
XXII
122
XXIII
124
XXIV
130
XXV
133
XXVI
135
XXVII
138
XXVIII
142
XXXVII
281
XXXVIII
303
XXXIX
341
XL
380
XLI
392
XLII
422
XLIII
452
XLIV
493
XLV
512
XLVI
552
XLVII
581
XLVIII
605
XLIX
621
L
625
LI
627
LII
631
LIII
637
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Page 45 - A graph is planar if and only if it contains no subdivision of...
Page 18 - A(n) is the number of partitions of n into an even number of distinct parts and B(n) is the number of partitions of n into an odd number of distinct parts.
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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