A complete resolution of the Keller maximum clique problem

Abstract

A d-dimensional Keller graph has vertices which are numbered with each of the 4 d possible d-digit numbers (d-tuples) which have each digit equal to 0, 1, 2, or 3. Two vertices are adjacent if their labels differ in at least two positions, and in at least one position the difference in the labels is two modulo four. Keller graphs are in the benchmark set of clique problems from the DIMACS clique challenge, and they appear to be especially difficult for clique algorithms. The dimension seven case was the last remaining Keller graph for which the maximum clique order was not known. It has been claimed in order to resolve this last case it might take a "high speed computer the size of a major galaxy". This paper describes the computation we used to determine that the maximum clique order for dimension seven is 124.

Properties, proved and conjectured, of Keller, Mycielski, and queen graphs

“…Give any other vertex (v i ) the same color as 2 vi 2 ; the "differ by 2" condition is never satisfied by both vertices (v i ) and 2 vi 2 . Therefore χ(G d ) ≤ 2 d (proved independently by Fung [11] and Debroni et al [6]). This coloring is also implicit in the proof of Theorem 6.4 below: the 0-and-2 set is the diagonal of the array shown.…”

“…The Keller graph G d of dimension d is defined as follows [6,21]: the 4 d vertices are all d-tuples from {0, 1, 2, 3}. Two tuples form an edge if they differ in at least two coordinates and if in at least one coordinate the difference of the entries is 2 (mod 4).…”

“…The Keller graphs play a critical role in the Keller conjecture [6], which, in its unrestricted form, states that any tiling of R d by unit cubes contains two cubes that meet face-to-face. This conjecture is closely related to ω( Table 2; see [6]). These ω values imply that the Keller conjecture with the restriction that all cube centers involve only integers or half-integers is true for d ≤ 7 and false for d ≥ 8.…”

On Maximum Weight Clique Algorithms, and How They Are Evaluated

“…The "hamming" and "johnson" graphs model problems from coding theory [7]. The "keller" instances encode a geometric conjecture [17], and the MANN family is made from clique formulations of the Steiner triple problem [36]. In each of these cases, the size of the solution has a real-world interpretation (and sometimes the vertices contained therein also convey meaning).…”

Maximum independent sets of the 120-cell and other regular polytopes

“…Both surveys cite the problem of finding a maximum clique of the Keller graph of dimension 7 as a open problem. This problem has subsequently been solved [5].…”