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 [math]\displaystyle{ e }[/math] is the shortness exponent of a graph family [math]\displaystyle{ {\mathcal F} }[/math], then every [math]\displaystyle{ n }[/math]-vertex graph in the family has a cycle of length near [math]\displaystyle{ n^e }[/math] but some graphs do not have longer cycles. More precisely, for any ordering of the graphs in [math]\displaystyle{ {\mathcal F} }[/math] into a sequence [math]\displaystyle{ G_0, G_1, \dots }[/math], with [math]\displaystyle{ h(G) }[/math] defined to be the length of the longest cycle in graph [math]\displaystyle{ G }[/math], the shortness exponent is defined as^[1]

[math]\displaystyle{ \liminf_{i\to\infty} \frac{\log h(G_i)}{\log |V(G_i)|}. }[/math]

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 [math]\displaystyle{ \log_3 2\approx 0.631 }[/math]. A construction based on kleetopes shows that some polyhedral graphs have longest cycle length [math]\displaystyle{ O(n^{\log_3 2}) }[/math],^[2] while it has also been proven that every polyhedral graph contains a cycle of length [math]\displaystyle{ \Omega(n^{\log_3 2}) }[/math].^[3] 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 [math]\displaystyle{ K_{2,n} }[/math]) with shortness exponent 0. There are many additional known results on shortness exponents of restricted subclasses of planar and polyhedral graphs.^[1]

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.^[4][5]

References

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