By Alison M. Marr

Magic squares are one of the extra renowned mathematical recreations. over the past 50 years, many generalizations of “magic” rules were utilized to graphs. lately there was a resurgence of curiosity in “magic labelings” as a result of a couple of effects that experience purposes to the matter of decomposing graphs into timber.

Key good points of this moment variation include:

·         a brand new bankruptcy on magic labeling of directed graphs

·         purposes of theorems from graph thought and engaging counting arguments

·         new learn difficulties and workouts masking a number of difficulties

·         an absolutely up-to-date bibliography and index

This concise, self-contained exposition is exclusive in its specialise in the speculation of magic graphs/labelings. it will possibly function a graduate or complicated undergraduate textual content for classes in arithmetic or computing device technology, and as reference for the researcher.

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Fifty three 2. 10 common labelings of C5 and C9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . sixty four 2. eleven Deficiency 6 on left; magic on correct . . . . . . . . . . . . . . . . . . . . . . . . . . sixty nine three. 1 A vertex-magic overall labeling of K4 − e . . . . . . . . . . . . . . . . . . . . . . seventy two three. 2 Vertex-magic overall labelings of W4 with an identical vertex-labels . . . seventy three three. three answer of the olympic earrings challenge . . . . . . . . . . . . . . . . . . . . . . . . seventy six xv xvi record of Figures three. four A twofold wheel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . seventy nine three. five Vertex-magic overall labeling of (t − 1)K1,3 ∪ K1,2 . . . . . . . . . . . . . . ninety two three. 6 Theorem three. 18 is healthier attainable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ninety eight four. 1 recognized absolutely magic graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 four. 2 Configurations forbidden by way of Theorem four. 6 . . . . . . . . . . . . . . . . . . . . . 116 four. three Configurations forbidden by way of Theorem four. 7 . . . . . . . . . . . . . . . . . . . . . 117 four. four Configurations forbidden via Theorem four. eight . . . . . . . . . . . . . . . . . . . . . 117 four. five A graph to be eradicated utilizing Theorem four. nine . . . . . . . . . . . . . . . . . . . a hundred twenty five four. 6 The graph G1200 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 four. 7 a unconditionally magic injection for G1200 . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 five. 1 smallest magic digraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 five. 2 9 non-magic digraphs on five vertices and 10 edges . . . . . . . . . . . . 138 five. three Magic labeling of DC5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . one hundred forty five. four Magic DC5 + C5 and DC6 + 2C3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 five. five A in the community magic digraph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 five. 6 Conservative labeling of K4,6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 five. 7 instance of an arc-magic labeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 five. eight Arc-magic tree labeling in strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 five. nine (a) An in-magic digraph, (b) an out-magic digraph, and (c) an in- and out-magic digraph . . . . . . . . . . . . . . . . . . . . . . . . . 153 five. 10 (a) Digraphs which are isomorphic after reversing edges and (b) digraphs that aren't isomorphic after reversing edges . . . . 154 1 Preliminaries 1. 1 Magic 1. 1. 1 Magic Squares Magic squares are one of the extra renowned mathematical recreations. Their origins are misplaced in antiquity. A classical reference is [2], whereas one of many higher contemporary books is [12]. A magic sq. of aspect n is an n× n array whose entries are an association of the integers {1, 2, . . . , n2 }, within which all components in any row, any column, or both the most diagonal or major back-diagonal, upload to a similar sum. Examples contain 1 15 eight 10 12 6 thirteen three 14 four eleven five 7 nine 2 sixteen 1 7 thirteen 19 25 18 24 five 6 12 10 eleven 17 23 four 22 three nine 15 sixteen 14 20 21 2 eight 34 6 five 32 1 33 nine 25 eleven eight 30 28 15 18 20 23 19 sixteen 22 24 14 17 thirteen 21 27 7 26 29 12 10 four 31 35 2 36 three adaptations within the set of entries have often been studied—for instance, one may well ask that the entries all be primes, or all be ideal squares—but we will A. M. Marr and W. D. Wallis, Magic Graphs, DOI 10. 1007/978-0-8176-8391-7 1, © Springer Science+Business Media manhattan 2013 1 2 1 Preliminaries basically have to speak about situations within which the entries are the 1st n2 confident integers. actually, we often should not have the fidelity of the diagonal and back-diagonal.

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