Golden-mean lower bound for all infinite transitive cubic graphs

Determine whether every connected, infinite, vertex-transitive, simple cubic graph G has connective constant μ(G) at least the golden mean φ=(1+√5)/2.

Background

The paper studies the connective constant μ(G), the exponential growth rate of self-avoiding walks on an infinite transitive graph. It establishes the inequality μ(G)≥φ for the 3-generator Cayley graphs of general Grigorchuk groups, with strict inequality under additional conditions on the defining sequence.

The broader class of connected, infinite, vertex-transitive, simple cubic graphs is denoted by the paper’s class of degree-3 transitive graphs. Although the golden-mean lower bound is proved for several subclasses, including two-ended cubic graphs and the Grigorchuk-group Cayley graphs considered in the paper, the unrestricted assertion for all such cubic graphs remains unresolved.