Shortness exponent

In graph theory, the shortness exponent is a numerical parameter of a family of graphs that measures how far from Hamiltonian the graphs in the family can be. Intuitively, if $$e$$ is the shortness exponent of a graph family $${\mathcal F}$$, then every $$n$$-vertex graph in the family has a cycle of length near $$n^e$$ but some graphs do not have longer cycles. More precisely, for any ordering of the graphs in $${\mathcal F}$$ into a sequence $$G_0, G_1, \dots$$, with $$h(G)$$ defined to be the length of the longest cycle in graph $$G$$, the shortness exponent is defined as
 * $$\liminf_{i\to\infty} \frac{\log h(G_i)}{\log |V(G_i)|}.$$

This number is always in the interval from 0 to 1; it is 1 for families of graphs that always contain a Hamiltonian or near-Hamiltonian cycle, and 0 for families of graphs in which the longest cycle length can be smaller than any constant power of the number of vertices.

The shortness exponent of the polyhedral graphs is $$\log_3 2\approx 0.631$$. A construction based on kleetopes shows that some polyhedral graphs have longest cycle length $$O(n^{\log_3 2})$$, while it has also been proven that every polyhedral graph contains a cycle of length $$\Omega(n^{\log_3 2})$$. The polyhedral graphs are the graphs that are simultaneously planar and 3-vertex-connected; the assumption of 3-vertex-connectivity is necessary for these results, as there exist sets of 2-vertex-connected planar graphs (such as the complete bipartite graphs $$K_{2,n}$$) with shortness exponent 0. There are many additional known results on shortness exponents of restricted subclasses of planar and polyhedral graphs.

The 3-vertex-connected cubic graphs (without the restriction that they be planar) also have a shortness exponent that has been proven to lie strictly between 0 and 1.