On Rigid Undirected Graphs

Abstract

By an undirected graph we mean a couple (X, R), where X is a set and R is a subset of X × X such that (x, y) ∈ R implies (y, x) ∈ R. The cardinal of X, denoted by |X|, will be called the cardinal of the graph.A mapping f:X → X is called an endomorphism of (X, R) if (x, y) ∈ R implies that (f(x), f(y)) ∈ R for all x, y ∈ R.An undirected graph (X, R) is called rigid if there is only one endomorphism of (X, R), namely the identity mapping of X.P. Erdös communicated orally that, using probability methods, it is possible to prove that almost all finite undirected graphs are rigid.