The colorings that improve the lower bounds for R(5,t), 10 <= t <= 15, are constructed by the same method. In each case we begin with the adjacency matrix B of a (3,t-1)-coloring. and let C be the adjacency matrix of the coloring complementary to B. Both B and C have color 1 on the main diagonal. Each of the (5,t)-colorings has an adjacency matrix with the following form (where we use X' to denote X-transpose):

B C B B C' B B B B' B' B C B' B' C' BThe complete the constructions, we need to specify the (3,t-1)-colorings that we are using to determine B. These are all circle colorings and are given in the table below.

t | Number of vertices | (3,t-1)-coloring used |
---|---|---|

10 | 140 | 35(1,7,11,16) |

11 | 152 | 38(1,4,11,13,19) |

12 | 180 | 45(3,10,11,12,16) |

13 | 192 | 48(5,6,8,15,22,24) |

14 | 220 | 55(3,7,11,12,13,27) |

15 | 236 | 59(3,9,11,15,16,21) |

Geoff Exoo