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Asymmetric graphs are graphs that have a trivial automorphism group $\textrm{Aut}(G)$, i.e. the only graph isomorphism from $G$ to itself is the identity.

Let's call a graph $G$ strongly asymmetric if the only graph homomorphism $h: G\to G$ is the identity (in other words, the endomorphism monoid $\textrm{End}(G)$ is trivial).

Given a (finite or infinite) cardinal $\kappa > 0$, is there a strongly asymmetric graph on $\kappa$ vertices?

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I believe the common name for such graphs is rigid. In fact, most random graphs are rigid. See this reference: On the minimal order of a graphs within a semigroup.

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  • $\begingroup$ A cautionary note: in many contexts, "rigid" is taken to mean merely "no nontrivial automorphisms" e.g. here: arxiv.org/pdf/math/9411236v1.pdf $\endgroup$ – Noah Schweber Nov 25 '14 at 17:33
  • $\begingroup$ That's right -- I think the reference given in the answer above might refer to existence of a trivial auto (but not endo) morphism only... I'm not sure we already have an answer for the question whether for any cardinal there is a rigid graph (in the strong sense). $\endgroup$ – Dominic van der Zypen Nov 25 '14 at 20:53

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