## Convexity and Graph TheoryAmong the participants discussing recent trends in their respective fields and in areas of common interest in these proceedings are such world-famous geometers as H.S.M. Coxeter, L. Danzer, D.G. Larman and J.M. Wills, and equally famous graph-theorists B. Bollobás, P. Erdös and F. Harary. In addition to new results in both geometry and graph theory, this work includes articles involving both of these two fields, for instance ``Convexity, Graph Theory and Non-Negative Matrices'', ``Weakly Saturated Graphs are Rigid'', and many more. The volume covers a broad spectrum of topics in graph theory, geometry, convexity, and combinatorics. The book closes with a number of abstracts and a collection of open problems raised during the conference. |

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### Contents

1 | |

13 | |

25 | |

Chapter 4 Some conditions for digraphs to be Hamiltonian | 37 |

Chapter 5 Covering of vertices of a simple graph with given connectivity and stability number | 43 |

Chapter 6 Embedding Latin squares in Steiner quasigroups and Howell designs in triple systems | 47 |

Chapter 7 Convexity graph theory and nonnegative matrices | 55 |

Chapter 8 Lattice points and lattice polytopes | 61 |

Chapter 26 Additive sequences of permutations | 197 |

Chapter 27 On pairs of disjoint segments in convex position in the plane | 203 |

Chapter 28 The decomposition of the nsphere and the boundaries of plane convex domains | 209 |

Chapter 29 Bounding the numbers of faces of polytope pairs and simple polyhedra | 215 |

Chapter 30 Parity in the realm of infinite permutations | 233 |

Chapter 31 NonHamiltonian simple 3polytopes with only one type of face besides triangles | 241 |

Chapter 32 At most 2d+1 neighborly simplices in Ed | 253 |

Chapter 33 Some recent results on cycle traversability in graphs | 255 |

Chapter 9 A new upper bound for the cardinality of 2distance sets in Euclidean space | 65 |

Chapter 10 Geodesics in oriented graphs | 67 |

Chapter 11 Recent results on the Strong Perfect Graph Conjecture | 75 |

Chapter 12 Iterative methods for the convex feasibility problem | 83 |

Chapter 13 Inequalities involving convex sets and their chords | 93 |

Chapter 14 A symmetrical arrangement of eleven hemiicosahedra | 103 |

Chapter 15 Regular incidencecomplexes and dimensionally unbounded sequences of such I | 115 |

Chapter 16 Some old and new problems in combinatorial geometry | 129 |

Chapter 17 PseudoBoolean functions and their graphs | 137 |

Chapter 18 Valuations of trees polynomial equations and polytopes | 147 |

Chapter 19 Spectra of trees | 151 |

Chapter 20 The valencefunctional | 161 |

Chapter 21 Matroids with weighted bases and Feynman integrals | 165 |

Chapter 22 Balanced pairs | 177 |

Chapter 23 On automorphisms of graphs with forbidden subgraphs | 183 |

Chapter 24 Weakly saturated graphs are rigid | 189 |

Chapter 25 Decomposability of polytopes is a projective invariant | 191 |

Chapter 34 Generalized Hamiltonian paths in tournaments | 263 |

Chapter 35 Large regular factors in random graphs | 271 |

Chapter 36 Techniques for investigating neighborly polytopes | 283 |

Chapter 37 Exchange properties of convexity spaces | 293 |

Hensleys lattice polytope theorem | 307 |

Chapter 39 Intersecting diameters in convex bodies | 311 |

Chapter 40 Quadratic forms and graphs Abstract | 317 |

Chapter 41 Approximation of convex bodies by polytopes with uniformly bounded valences Abstract | 319 |

Chapter 42 Some results on convexity in graphs Abstract | 321 |

Achievement and avoidance games Abstract | 323 |

Chapter 44 On scheduling the construction of a tree Abstract | 325 |

Chapter 45 Edge maps and isomorphisms of graphs Abstract | 327 |

Chapter 46 A multiflow problem without the maxflow mincut property Abstract | 329 |

Chapter 47 Polyhedral manifolds with few vertices Abstract | 331 |

Chapter 48 Regular polyhedral manifolds Abstract | 333 |

Open problems | 335 |

### Common terms and phrases

1981 North-Holland 3-connected additive sequence adjacent assume automorphism binary sequence chords Combinat combinatorial complete cone conjecture connected convex sets convexity space Corollary corresponding curve cyclic deﬁned deﬁnition diameter Discr disjoint elements equivalent Euclidean example exchange function exchange number exists extreme rays face of Q facets ﬁnd ﬁnitary ﬁnite ﬁrst ﬁxed points geometric given graph G Graph Theory Griinbaum H.S.M. Coxeter Hamiltonian paths hemi-icosahedron Hence holds hyperplane hypohamiltonian graph implies inequalities inﬁnite integer isomorphic Israel Latin square lattice Left Blank North-Holland Lemma Let G linear lower bound mapping Math Mathematics matrix matroid n-connected neighborly 2m-polytope non-Hamiltonian non-negative number of edges Paley graph partitionable Perfect Graph permutations planar graphs polyhedral polynomial polytope pair proof of Theorem properties prove quasigroup realizations regular graphs result satisﬁes Section simplicial d-polytope Steiner quasigroup Steiner triple systems subgraph subset Theorem tournament tree triangle University upper bound vector vertex vertices