 
Summary: COMBINATOKICA6 (3) (1986) 201206
COVERING GRAPHS BY THE MINIMUM
NUMBER OF EQUIVALENCE RELATIONS
N. ALON
Received15 December1984
An equivalence graph is a vertex disjoint union of complete graphs. For a graph G, let
eq(G) be the irdnimum number of equivalence subgraphs of G needed to cover all edges of G. Simi
larly, let cc(G) be the minimum number of complete subgraphs of G needed to cover all its edges.
Let H be a graph on n vertices with ma,'dmal degree _~d(and minimal degree ~1), and let G=I~
be its complement. We show that
log2nlog2d ~_eq(G) ~ cc (G) ~_ 2e2(d+ 1)~logan.
The lower bound is proved by multilinear techniques (exterior algebra), and its assertion for the
complement of an ncycle settles a problem of Frankl. The upper bound is proved by probabilistie
arguments, and it generalizes results of de Caen, Gregory and Pullman.
1. Introduction
All graphs considered here are finite, simple and undirected. Let V be a finite
set. For an equivalence relation R on V, let G(R) denote its graph, i.e., the graph
on Vin which x, yC V are adjacent iff x is in relation with y. We call G(R)an equ
ivalence graph. Clearly a ~aph is an equivalence graph iff it is a vertex disjoint union
of complete graphs. An equivalence covering of a graph G is a family of equivalence
