COMMON MULTIPLES OF COMPLETE GRAPHS

Abstract

A graph $H$ is said to divide a graph $G$ if there exists a set $S$ of subgraphs of $G$, all isomorphic to $H$, such that the edge set of $G$ is partitioned by the edge sets of the subgraphs in $S$. Thus, a graph $G$ is a common multiple of two graphs if each of the two graphs divides $G$.

This paper considers common multiples of a complete graph of order $m$ and a complete graph of order $n$. The complete graph of order $n$ is denoted $K_n$. In particular, for all positive integers $n$, the set of integers $q$ for which there exists a common multiple of $K_3$ and $K_n$ having precisely $q$ edges is determined.

It is shown that there exists a common multiple of $K_3$ and $K_n$ having $q$ edges if and only if $q \equiv 0 \, ({\rm mod}\, 3)$, $q \equiv 0 \, ({\rm mod}\, \binom n2)$ and

(1) $q \neq 3 \binom n2$ when $n \equiv 5 \, ({\rm mod}\, 6)$;

(2) $q \geq (n + 1) \binom n2$ when $n$ is even;

(3) $q \notin \{36, 42, 48\}$ when $n = 4$.

The proof of this result uses a variety of techniques including the use of Johnson graphs, Skolem and Langford sequences, and equitable partial Steiner triple systems.

2000 Mathematical Subject Classification: 05C70, 05B30, 05B07.