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Packing of graphs

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CONTENTS Preface............................................................................5 1. Introduction.................................................................5   1.1. Basic graph-theoretic terms...................................6   1.2. Some families of graphs.........................................8   1.3. Edge-disjoint placements of graphs.......................9 2. Embeddings of graphs................................................9   2.1. Basic result............................................................9   2.2. Self-complementary permutations........................10   2.3. Embeddings without fixed points...........................15   2.4. Graphs without small cycles..................................18   2.5. Uniquely embeddable graphs...............................23 3. Packing of two graphs...............................................23   3.1. Packing of two graphs of small size......................23   3.2. Packing an undense and a dense graph.............25   3.3. Products of sizes and degrees.............................26   3.4. Sum of sizes.........................................................28   3.5. Erdős-Sós Conjecture..........................................31     3.5.1. Special families of trees...................................31     3.5.2. Particular values of parameters.......................37     3.5.3. Special families of graphs................................39   3.6. Other problems related to trees and forests.........40   3.7. Some generalizations...........................................41 4. Packing of three graphs............................................45   4.1. Triple placement of graphs...................................45   4.2. Permutation structure...........................................50   4.3. 3-placement of a tree...........................................52   4.4. Packing three trees..............................................54   4.5. Packing three trees - general case......................58   4.6. Packing three forests...........................................58 5. Some special problems.............................................59   5.1. Packing a graph with its square...........................59   5.2. Careful packing of a graph...................................62   5.3. Packing of sequences of trees.............................66     5.3.1. Tree Packing Conjecture.................................66     5.3.2. Not too large trees...........................................69   5.4. Bipartite graphs....................................................70   5.5. Packing of digraphs..............................................72 Bibliography...................................................................75
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