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 ed