  The notion of a competition graph was introduced by J. E. Cohen in 1968. The
competition graph C(D) of a digraph D is a (simple undirected) graph which has
the same vertex set as D and has an edge between two distinct vertices x and y
if and only if there exists a vertex v in D such that (x,v) and (y,v) are arcs
of D. For any graph G, G together with sufficiently many new isolated vertices
is the competition graph of some acyclic digraph. In 1978, F. S. Roberts
defined the competition number k(G) of a graph G as the minimum number of such
isolated vertices. In general, it is hard to compute the competition number
k(G) for a graph G and it has been one of important research problems in the
study of competition graphs to characterize a graph by computing its
competition number.
  It is well known that there exist 5 kinds of regular polyhedra in the three
dimensional Euclidean space: a tetrahedron, a hexahedron, an octahedron, a
dodecahedron, and an icosahedron. We regard a polyhedron as a graph. The
competition numbers of a tetrahedron, a hexahedron, an octahedron, and a
dodecahedron are easily computed by using known results on competition numbers.
In this paper, we focus on an icosahedron and give the exact value of the
competition number of an icosahedron.
