On a Problem of Cameron’s on Inexhaustible Graphs

A graph G is inexhaustible if whenever a vertex of G is deleted the remaining graph is isomorphic to G. We address a question of Cameron [6], who asked which countable graphs are inexhaustible. In particular, we prove that there are continuum many countable inexhaustible graphs with properties in common with the infinite random graph, including adjacency properties and universality. Locally finite inexhaustible graphs and forests are investigated, as is a semigroup structure on the class of inexhaustible graphs. We extend a result of [7] on homogeneous inexhaustible graphs to pseudo-homogeneous inexhaustible graphs.

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Correspondence to Anthony Bonato.

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The authors gratefully acknowledge support from the Natural Science and Engineering Research Council of Canada (NSERC).

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Bonato, A., Delić, D. On a Problem of Cameron’s on Inexhaustible Graphs. Combinatorica 24, 35–51 (2004). https://doi.org/10.1007/s00493-004-0003-1

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Mathematics Subject Classification (2000):

  • 05C75
  • 03C15