On the factorisation of the complete graph into factors of diameter 2

Abstract

f(k) denotes the smallest number n such that the complete graph (n) can be decomposed into k factors of diameter 2. So far the following results have been obtained [1]:

$4k-1⩽f\left(k\right)⩽\left(\begin{array}{c}6k-7\\ 2k-2\end{array}\right)$

f(2)≤5, f(3)≤13, f(4)≤41, f(5)≤71, f(6)≤157, f(7)≤193, f(8)≤193, f(9)≤379, f(10)≤521 and there exists a positive integer K such that for any integer k>K:

$f\left(k\right)⩽{\left(\frac{49}{10}\right)}^2{k}^{2}logk$

The purpose of this paper is to improve the upper bound on f(k) by showing that f(k)≤7k holds.