# Girth (graph theory)

In graph theory, the **girth** of a graph is the length of a shortest cycle contained in the graph.[1] If the graph does not contain any cycles (i.e. it's an acyclic graph), its girth is defined to be infinity.[2]
For example, a 4-cycle (square) has girth 4. A grid has girth 4 as well, and a triangular mesh has girth 3. A graph with girth four or more is triangle-free.

## Cages

A cubic graph (all vertices have degree three) of girth *g* that is as small as possible is known as a *g*-cage (or as a (3,*g*)-cage). The Petersen graph is the unique 5-cage (it is the smallest cubic graph of girth 5), the Heawood graph is the unique 6-cage, the McGee graph is the unique 7-cage and the Tutte eight cage is the unique 8-cage.[3] There may exist multiple cages for a given girth. For instance there are three nonisomorphic 10-cages, each with 70 vertices: the Balaban 10-cage, the Harries graph and the Harries–Wong graph.

## Girth and graph coloring

For any positive integers *g* and χ, there exists a graph with girth at least *g* and chromatic number at least χ; for instance, the Grötzsch graph is triangle-free and has chromatic number 4, and repeating the Mycielskian construction used to form the Grötzsch graph produces triangle-free graphs of arbitrarily large chromatic number. Paul Erdős was the first to prove the general result, using the probabilistic method.[4] More precisely, he showed that a random graph on *n* vertices, formed by choosing independently whether to include each edge with probability *n*^{(1 − g)/g}, has, with probability tending to 1 as *n* goes to infinity, at most *n*/2 cycles of length *g* or less, but has no independent set of size *n*/2*k*. Therefore, removing one vertex from each short cycle leaves a smaller graph with girth greater than *g*, in which each color class of a coloring must be small and which therefore requires at least *k* colors in any coloring.

Explicit, though large, graphs with high girth and chromatic number can be constructed as certain Cayley graphs of linear groups over finite fields.[5] These remarkable *Ramanujan graphs* also have large expansion coefficient.

## Related concepts

The **odd girth** and **even girth** of a graph are the lengths of a shortest odd cycle and shortest even cycle respectively.

The **circumference** of a graph is the length of the *longest* cycle, rather than the shortest.

Thought of as the least length of a non-trivial cycle, the girth admits natural generalisations as the 1-systole or higher systoles in systolic geometry.

Girth is the dual concept to edge connectivity, in the sense that the girth of a planar graph is the edge connectivity of its dual graph, and vice versa. These concepts are unified in matroid theory by the girth of a matroid, the size of the smallest dependent set in the matroid. For a graphic matroid, the matroid girth equals the girth of the underlying graph, while for a co-graphic matroid it equals the edge connectivity.[6]

## References

- R. Diestel,
*Graph Theory*, p.8. 3rd Edition, Springer-Verlag, 2005 -
*Girth – Wolfram MathWorld* - Brouwer, Andries E.,
*Cages*. Electronic supplement to the book*Distance-Regular Graphs*(Brouwer, Cohen, and Neumaier 1989, Springer-Verlag). - Erdős, Paul (1959), "Graph theory and probability",
*Canadian Journal of Mathematics*,**11**: 34–38, doi:10.4153/CJM-1959-003-9. - Guiliana Davidoff, Peter Sarnak, Alain Valette,
*Elementary number theory, group theory, and Ramanujan graphs*, Cambridge University Press, 2003. - Cho, Jung Jin; Chen, Yong; Ding, Yu (2007), "On the (co)girth of a connected matroid",
*Discrete Applied Mathematics*,**155**(18): 2456–2470, doi:10.1016/j.dam.2007.06.015, MR 2365057.