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shortnesses

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Definitions of shortnesses in various dictionaries:

noun - the property of being of short spatial extent

noun - the condition of being short of something

noun - the property of being truncated or short

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Shortnesses might refer to
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 * * * {\displaystyle e} * is the shortness exponent of a graph family * * * * * * F * * * * * {\displaystyle {\mathcal {F}}} * , then every * * * * n * * * {\displaystyle n} * -vertex graph in the family has a cycle of length near * * * * * n * * e * * * * * {\displaystyle n^{e}} * but some graphs do not have longer cycles. More precisely, for any ordering of the graphs in * * * * * * F * * * * * {\displaystyle {\mathcal {F}}} * into a sequence * * * * * G * * 0 * * * , * * G * * 1 * * * , * … * * * {\displaystyle G_{0},G_{1},\dots } * , with * * * * h * ( * G * ) * * * {\displaystyle h(G)} * defined to be the length of the longest cycle in graph * * * * G * * * {\displaystyle G} * , the shortness exponent is defined as* * * * * lim inf * * i * → * ∞ * * * * * * log * ⁡ * h * ( * * G * * i * * * ) * * * log * ⁡ * * \| * * V * ( * * G * * i * * * ) * * \| * * * * * . * * * {\displaystyle \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 * ≈ * 0.631 * * * {\displaystyle \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 * * * ...

Shortnesses might refer to

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
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* {\displaystyle \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
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