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On Rigid Undirected Graphs

Published online by Cambridge University Press:  20 November 2018

Z. Hedrlín  and
A. Pultr
Z. Hedrlín
Affiliation:
Caroline University, Prague
A. Pultr
Affiliation:
Caroline University, Prague
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 1237 - 1242
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Hedrlin, Z. and Pultr, A., Symmetric relations (undirected graphs) with given semigroups, Monatsh. Math., 69 (1965), 318–322.Google Scholar
2. Kagno, I. N., Linear graphs of degree ≤6 and their groups, Amer. J. Math., 68 (1946), 505–520.Google Scholar
3. Vopĕnka, P., Pultr, A., and Hedrlín, Z., A rigid relation exists on any set, Comment. Math. Univ. Carolinae, 6 (1965), 149–155.Google Scholar