## Arranging points in Rd̳: a question of balance |

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### Contents

Notation and remarks | 21 |

Two graph theory problems as special cases | 64 |

Variations on the balancing problem | 80 |

1 other sections not shown

### Common terms and phrases

3-circuits algorithm balancing problem bound 2d C(Kr chromatic number problem cocircuit space color column vectors Combinatorial Theory complete graph completes the proof configuration in Rd connected components consider contains the origin conv(A convex hull Corollary corresponding d-dimensional decomposition defined Denote directed circuit directed graph e(pj elementary vectors equivalent Euclidean example finite nondegenerate configuration finite set finite subset G is k-colorable graph G graph theory H e H0 Hence hyperplane H incidence matrix incidence vector inequality k-colorable Larman Lemma Let h linear spaces linear subspace linearly independent nondegenerate linear nonzero vector norm NP-complete ordering orientation of G oriented matroid orthogonal partial sums perfect matching polynomial positive integer procedure proof of Theorem real number row space row vectors S+(x satisfies signed sets Spencer strongly connected components subspace of Rn Suppose supremum totally unimodular matrix two-coloring vertex vertex-edge incidence matrix vertices