Girth (graph theory)

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.

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 circumference 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.

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.

Computation

The girth of an undirected graph can be computed by running a breadth-first search from each node, with complexity ${\displaystyle O(nm)}$ where ${\displaystyle n}$ is the number of vertices of the graph and ${\displaystyle m}$ is the number of edges. A practical optimization is to limit the depth of the BFS to a depth that depends on the length of the smallest cycle discovered so far. Better algorithms are known in the case where the girth is even and when the graph is planar. In terms of lower bounds, computing the girth of a graph is at least as hard as solving the triangle finding problem on the graph.