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arXiv · 2503.11798

Elusive properties of countably infinite graphs

Márton Elekes·Tamás Kátay·Anett Kocsis
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Abstract

A graph property is elusive (or evasive) if any algorithm testing it by asking questions of the form ''Is there an edge between vertices x and y?'' must, in the worst case, examine all pairs of vertices. Elusiveness for infinite vertex sets has been first studied by Csern\'ak and Soukup, who proved that the long-standing Aanderaa-Karp-Rosenberg Conjecture -- which states that every nontrivial monotone graph property is elusive -- fails for infinite vertex sets. We extend their work by giving a closer look to the case when the vertex set is countably infinite and the ''algorithm'' terminates after infinitely many steps. Among others, we prove that connectedness is elusive, which strengthens a result of Csern\'ak and Soukup. We give counterexamples to the infinite version of the Aanderaa-Karp-Rosenberg Conjecture even if the ''algorithm'' is required to terminate after infinitely many steps, which strengthens results of Csern\'ak and Soukup.

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Márton Elekes, Tamás Kátay, Anett Kocsis. 2025-03-14. Elusive properties of countably infinite graphs. https://arxiv.org/abs/2503.11798

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