In graph theory, the girth of an undirected graph is the length of a shortest

cycle Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in soc ...

contained in the graph. If the graph does not contain any cycles (that is, it is a forest), its girth is defined to be

infinity Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...

. 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 In the mathematical area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle of edges. Triangle-free graphs may be equivalently defined as graphs with clique number ≤ 2, graphs with g ...

Cages

A cubic graph (all vertices have degree three) of girth that is as small as possible is known as a - cage (or as a -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. 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 In the mathematical field of graph theory, the Balaban 10-cage or Balaban -cage is a 3-regular graph with 70 vertices and 105 edges named after Alexandru T. Balaban. Published in 1972, It was the first 10-cage discovered but it is not unique. ...

, the

Harries graph In the mathematical field of graph theory, the Harries graph or Harries (3-10)-cage is a 3- regular, undirected graph with 70 vertices and 105 edges. The Harries graph has chromatic number 2, chromatic index 3, radius 6, diameter 6, girth 10 an ...

and the

Harries–Wong graph In the mathematical field of graph theory, the Harries–Wong graph is a 3- regular undirected graph with 70 vertices and 105 edges. The Harries–Wong graph has chromatic number 2, chromatic index 3, radius 6, diameter 6, girth 10 and is ...

. Image:Petersen1 tiny.svg, The Petersen graph has a girth of 5 Image:Heawood_Graph.svg, The Heawood graph has a girth of 6 Image:McGee graph.svg, The McGee graph has a girth of 7 Image:Tutte eight cage.svg, The Tutte–Coxeter graph (''Tutte eight cage'') has a girth of 8

Girth and graph coloring

For any positive integers and , there exists a graph with girth at least and chromatic number at least ; 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 Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in ...

was the first to prove the general result, using the probabilistic method. More precisely, he showed that a random graph on vertices, formed by choosing independently whether to include each edge with probability , has, with probability tending to 1 as goes to infinity, at most cycles of length or less, but has no independent set of size . Therefore, removing one vertex from each short cycle leaves a smaller graph with girth greater than , in which each color class of a coloring must be small and which therefore requires at least 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. 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 of a graph is the length of the ''longest'' (simple) 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 In mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth, and others, ...

. Girth is the dual concept to

edge connectivity In graph theory, a connected graph is -edge-connected if it remains connected whenever fewer than edges are removed. The edge-connectivity of a graph is the largest for which the graph is -edge-connected. Edge connectivity and the enumeration ...

, in the sense that the girth of a

planar graph In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross ...

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.{{citation , last1 = Cho , first1 = Jung Jin , last2 = Chen , first2 = Yong , last3 = Ding , first3 = Yu , doi = 10.1016/j.dam.2007.06.015 , issue = 18 , journal = Discrete Applied Mathematics , mr = 2365057 , pages = 2456–2470 , title = On the (co)girth of a connected matroid , volume = 155 , year = 2007, doi-access = free .

References

Graph invariants