On the girth of infinite graphs

Abstract

Let X be an infinite k-valent graph with polynomial growth of degree d, i.e. there is an integer d and a constant c such that f x ( n)⩽ cn d , where f x ( n) denotes the growth function of X. We show that the girth of such a graph has an upper bound depending on c and d. We also prove that for every triple ( k, d, l) of integers with k⩾3, d⩾1, l⩾3, there exist k-valent connected graphs with polynomial growth of degree d and girth greater than l. This means that in general the girth of graphs with polynomial growth mainly depends on the constant c in the upper bound of the growth function. We also prove lower bounds for the girth of Cayley graphs of certain classes of groups with polynomial growth.