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Minimum covering by triples, quadruples and quintuples with minimum excess

Petr Kovář, Yifan Zhang

Abstract

This article explores a new type of optimal covering of a complete graph by small cliques of different sizes, namely the minimum covering with minimum excess. In particular, the minimum size of a covering by triples and quadruples with minimum excess is determined. Moreover, some generalisation onto minimum coverings by triples, quadruples and quintuples with minimum excess is presented.

Minimum covering by triples, quadruples and quintuples with minimum excess

Abstract

This article explores a new type of optimal covering of a complete graph by small cliques of different sizes, namely the minimum covering with minimum excess. In particular, the minimum size of a covering by triples and quadruples with minimum excess is determined. Moreover, some generalisation onto minimum coverings by triples, quadruples and quintuples with minimum excess is presented.
Paper Structure (21 sections, 77 theorems, 176 equations, 10 tables)

This paper contains 21 sections, 77 theorems, 176 equations, 10 tables.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Basic notations
  4. Block designs
  5. Main results
  6. Proofs of results
  7. Minimum excess in a $\{K_3,K_4\}$-cover
  8. Minimum $\{K_3,K_4\}$-cover with minimum excess
  9. Optimal designs for $v\equiv 1\pmod{3}$.
  10. Optimal designs for $v\equiv 0\pmod{3}$.
  11. Optimal designs for $v\equiv 2\pmod{3}$.
  12. Small exceptional cases
  13. Minimum excess in a $\{K_3,K_4,K_5\}$-cover
  14. Minimum $\{K_3,K_4,K_5\}$-cover with minimum excess
  15. Open cases
  16. ...and 6 more sections

Key Result

Theorem 1

Denote by then we have

Theorems & Definitions (140)

  • Theorem 1: Stanton1983Stanton1988Gruettmueller2006
  • Theorem 2: kirkman-1847
  • Theorem 3: hanani-1975, p. 295, Theorem 5.2
  • Theorem 4: hanani-1975, p. 302, Theorem 5.3
  • Theorem 5: Lu1990, RayChaudhuri1971
  • Theorem 6: handbook-designs, IV.1.24
  • Theorem 7: rees-1989
  • Theorem 8: handbook-designs, IV.1.27
  • Theorem 9: hanani-1975, p. 357, Theorem 6.2
  • Theorem 10: Colbourn1992
  • ...and 130 more