## Handbook of Combinatorics, Volume 1Handbook of Combinatorics, Volume 1 focuses on basic methods, paradigms, results, issues, and trends across the broad spectrum of combinatorics. The selection first elaborates on the basic graph theory, connectivity and network flows, and matchings and extensions. Discussions focus on stable sets and claw free graphs, nonbipartite matching, multicommodity flows and disjoint paths, minimum cost circulations and flows, special proof techniques for paths and circuits, and Hamilton paths and circuits in digraphs. The manuscript then examines coloring, stable sets, and perfect graphs and embeddings and minors. The book takes a look at random graphs, hypergraphs, partially ordered sets, and matroids. Topics include geometric lattices, structural properties, linear extensions and correlation, dimension and posets of bounded degree, hypergraphs and set systems, stability, transversals, and matchings, and phase transition. The manuscript also reviews the combinatorial number theory, point lattices, convex polytopes and related complexes, and extremal problems in combinatorial geometry. The selection is a valuable reference for researchers interested in combinatorics. |

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### Contents

University of Washington Seattle WA Ch | 18 |

Hamilton paths and circuits in graphs | 20 |

Hamilton paths and circuits in digraphs | 28 |

Lenstra J K Eindhoven University of Technology Eindhoven and Centre | 35 |

Lloyd E K University of Southampton Southampton Ch | 44 |

Fundamental classes of graphs and digraphs | 54 |

Automorphism Groups Isomorphism Reconstruction 1447 | 64 |

Special proof techniques for paths and circuits | 69 |

Finite Sets and Relations | 381 |

Probabilistic Methods 1785 | 385 |

Partially Ordered Sets | 433 |

Matroids | 481 |

Matroid Minors | 527 |

Matroid Optimization and Algorithms | 551 |

Symmetric Structures | 611 |

Finite Geometries | 647 |

Packings and coverings by paths and circuits | 80 |

References | 94 |

Combinatorial Optimization 1541 | 98 |

Connectivity and Network Flows | 111 |

References | 170 |

Computational Complexity 1599 | 173 |

Matchings and Extensions | 179 |

Tools from Linear Algebra 1705 | 222 |

Combinatorial Games 2117 | 229 |

Colouring Stable Sets and Perfect Graphs | 233 |

NowhereZero Flows | 289 |

Embeddings and Minors | 301 |

The History of Combinatorics 2163 | 308 |

Tools from Higher Algebra 1749 | 312 |

Random Graphs | 351 |

Block Designs | 693 |

Association Schemes | 747 |

Codes | 773 |

Combinatorial Structures in Geometry and Number Theory | 809 |

Convex Polytopes and Related Complexes | 875 |

Topological Methods 1819 | 896 |

Point Lattices | 919 |

Combinatorics in Computer Science 2003 | 961 |

Combinatorial Number Theory | 967 |

Combinatorics in Pure Mathematics 2039 | 1018 |

xiii | |

lix | |

xcii | |

### Common terms and phrases

3-connected algebraic algorithm Amer association scheme binary bipartite graph blocks called chapter chromatic number colour Combin combinatorial complete components Comput conjecture connected graph construction convex convex sets Corollary cycle d-polytope decomposition defined denote designs digraph dimension Discrete Math disjoint dual elements embedding equivalent Erdos example exists finite function G contains geometry given graph G Graph Theory Hamilton circuit hamiltonian hypergraph implies incident induced inequality integer intersection isomorphic lattice least Lemma Let G linear London Math Lovasz matrix matroid maximal maximum number minimal nodes nowhere-zero obtained oriented matroids pairs partition path perfect graph perfect matching permutation groups Petersen graph planar graph points polymatroid polynomial polytopes poset problem Proc projective plane proof proved random graph result simple graph simplicial stable set Steiner systems strongly regular graph subgraph submodular subset symmetric Theorem Thomassen tree Tutte vector vertex vertices