Combinatorics and Graph Theory

2307 Submissions

[2] viXra:2307.0058 [pdf] submitted on 2023-07-11 20:43:09

On Conway's 99-graph: Parts 1-20

Authors: E. Guseinov
Comments: 95 Pages.

This preprint in Russian reflects some of my results on Conway's 99-graph (G) obtained in 2022-2023 during a research that I did on my own. Here I prove that G is Cartesian indecomposable (Part 10, Theorem 36), G can not be obtained by a group-theoretic generalization of Berlekamp-Seidel's construction (Part 11), G contains no Hamming graph H(4,3) (Part 8, Theorem 28), G is not contained in strongly regular graphs srg(243,22,1,2) (Part 12, Theorem 40), the independence number of G is at least 10 (Part 4, Theorem 21). Parts 12-17, 19 and 20 reflect attempts to prove G may contain H(2,3), i. e. the Paley graph P(9). I'm currently accepting offers to join a graph theoretical research, so you can write me on elmarguseinov@yahoo.com.
Category: Combinatorics and Graph Theory

[1] viXra:2307.0008 [pdf] replaced on 2024-01-05 01:58:19

General Conjecture on the Optimal Covering Trails in a K-Dimensional Cubic Lattice

Authors: Marco Ripà
Comments: 7 Pages.

We introduce a general conjecture involving minimum-link covering trails for any given k-dimensional grid n × n × ··· × n, belonging to the cubic lattice ℕ^k. In detail, if n is above two, we hypothesize that the minimal link length of any covering trail, for the above-mentioned set of n^k points in the Euclidean space ℝ^k, is equal to h(n, k) = (n^k − 1)/(n − 1) + c·(n − 3), where c = k − 1 iff h(4, 3) = 23, c = 1 iff h(4, 3) = 22, or even c = 0 iff h(4, 3) = 21.
Category: Combinatorics and Graph Theory