Abstract
Apart from four trivial quasivarieties of bipartite graphs, any finitely axiomatizable universal Horn class of graphs must contain graphs of arbitrarily large chromatic number. Hence, no finitely generated universal Horn class of graphs is finitely axiomatizable, except these four. On the other hand, there is a continuum of universal Horn classes of graphs.
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Caicedo, X. Finitely axiomatizable quasivarieties of graphs. Algebra Universalis 34, 314–321 (1995). https://doi.org/10.1007/BF01204787
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DOI: https://doi.org/10.1007/BF01204787