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 4cycle (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 trianglefree.
Cages
A cubic graph (all vertices have degree three) of girth g that is as small as possible is known as a gcage (or as a (3,g)cage). The Petersen graph is the unique 5cage (it is the smallest cubic graph of girth 5), the Heawood graph is the unique 6cage, the McGee graph is the unique 7cage and the Tutte eight cage is the unique 8cage.^{[3]} There may exist multiple cages for a given girth. For instance there are three nonisomorphic 10cages, each with 70 vertices : the Balaban 10cage, the Harries graph and the Harries–Wong graph.

The Petersen graph has a girth of 5

The Heawood graph has a girth of 6

The McGee graph has a girth of 7

The Tutte–Coxeter graph (Tutte eight cage) has a girth of 8
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 trianglefree and has chromatic number 4, and repeating the Mycielskian construction used to form the Grötzsch graph produces trianglefree 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/2k. 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.
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 nontrivial cycle, the girth admits natural generalisations as the 1systole 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 cographic matroid it equals the edge connectivity.^{[5]}
References
 ↑ R. Diestel, Graph Theory, p.8. 3rd Edition, SpringerVerlag, 2005
 ↑ Girth – Wolfram MathWorld
 ↑ Brouwer, Andries E., Cages. Electronic supplement to the book DistanceRegular Graphs (Brouwer, Cohen, and Neumaier 1989, SpringerVerlag).
 ↑ Erdős, Paul (1959), "Graph theory and probability", Canadian Journal of Mathematics, 11: 34–38, doi:10.4153/CJM19590039.
 ↑ 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.