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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