竞争图的概念是由J.E. 1968年的科恩。有向图D的竞争图C（D）是一个（简单无向）图，它与D具有相同的顶点集，并且在两个不同的顶点x和y之间有一条边，当且仅当D中存在一个顶点v，使得（x，v）和（y，v）是D的弧。对任意图G，G加上足够多的新孤立点是某个无圈有向图的竞争图。1978年，F. S. Roberts将图G的竞争数k（G）定义为图G中孤立点的最小个数。一般来说，图G的竞争数k（G）是很难计算的，通过计算图G的竞争数来刻画图G是竞争图研究中的重要问题之一。众所周知，在三维欧几里得空间中存在5种正多面体：四面体、六面体、八面体、十二面体和二十面体。我们把多面体看作一个图.利用已知的竞争数结果，可以很容易地计算四面体、六面体、八面体和十二面体的竞争数。本文以二十面体为研究对象，给出了二十面体竞争数的精确值。
英文摘要：
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.
