Definitions

# Girth (graph theory)

In graph theory, the girth of a graph is the length of a shortest cycle contained in the graph. If the graph does not contain any cycles, its girth is defined to be infinity. 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 ≥ 4 is triangle-free.

## Cages

A cubic graph of girth $g$ that is as small as possible is known as a $g$-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, and the Tutte eight cage is the unique 8-cage.

## 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. 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.