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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 is the shortness exponent of a graph family , then every -vertex graph in the family has a cycle of length near but some graphs do not have longer cycles. More precisely, for any ordering of the graphs in into a sequence , with defined to be the length of the longest cycle in graph , the shortness exponent is defined as^{[1]}

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 . A construction based on kleetopes shows that some polyhedral graphs have longest cycle length ,^{[2]} while it has also been proven that every polyhedral graph contains a cycle of length .^{[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 ) 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]}

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^{a}^{b}Grünbaum, Branko; Walther, Hansjoachim (1973), "Shortness exponents of families of graphs",*Journal of Combinatorial Theory*, Series A,**14**: 364–385, doi:10.1016/0097-3165(73)90012-5, hdl:10338.dmlcz/101257, MR 0314691. **^**Moon, J. W.; Moser, L. (1963), "Simple paths on polyhedra",*Pacific Journal of Mathematics*,**13**: 629–631, doi:10.2140/pjm.1963.13.629, MR 0154276.**^**Chen, Guantao; Yu, Xingxing (2002), "Long cycles in 3-connected graphs",*Journal of Combinatorial Theory*, Series B,**86**(1): 80–99, doi:10.1006/jctb.2002.2113, MR 1930124.**^**Bondy, J. A.; Simonovits, M. (1980), "Longest cycles in 3-connected 3-regular graphs",*Canadian Journal of Mathematics*,**32**(4): 987–992, doi:10.4153/CJM-1980-076-2, MR 0590661.**^**Jackson, Bill (1986), "Longest cycles in 3-connected cubic graphs",*Journal of Combinatorial Theory*, Series B,**41**(1): 17–26, doi:10.1016/0095-8956(86)90024-9, MR 0854600.