girth (graph theory)

{{main|Cage (graph theory)}}

A cubic graph (all vertices have degree three) of girth {{mvar|g}} that is as small as possible is known as a {{mvar|g}}-cage (or as a {{math|(3,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, the McGee graph is the unique 7-cage and the Tutte eight cage is the unique 8-cage.{{citation|first=Andries E.|last=Brouwer|author-link=Andries Brouwer|url=http://www.win.tue.nl/~aeb/drg/graphs/|title=Cages}}. Electronic supplement to the book Distance-Regular Graphs (Brouwer, Cohen, and Neumaier 1989, Springer-Verlag). 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, the Harries graph and the Harries–Wong graph.

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

For any positive integers {{mvar|g}} and {{math|χ}}, there exists a graph with girth at least {{mvar|g}} and chromatic number at least {{math|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.{{citation | last = Erdős | first = Paul | author-link = Paul Erdős | journal = Canadian Journal of Mathematics | pages = 34–38 | title = Graph theory and probability | volume = 11 | year = 1959 | doi = 10.4153/CJM-1959-003-9| s2cid = 122784453 | doi-access = free }}. More precisely, he showed that a random graph on {{mvar|n}} vertices, formed by choosing independently whether to include each edge with probability {{math|n^(1–g)/g}}, has, with probability tending to 1 as {{mvar|n}} goes to infinity, at most {{math|{{sfrac|n|2}}}} cycles of length {{mvar|g}} or less, but has no independent set of size {{math|{{sfrac|n|2k }}}}. Therefore, removing one vertex from each short cycle leaves a smaller graph with girth greater than {{mvar|g}}, in which each color class of a coloring must be small and which therefore requires at least {{mvar|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.{{citation | last1 = Davidoff | first1 = Giuliana | author1-link = Giuliana Davidoff | last2 = Sarnak | first2 = Peter | author2-link = Peter Sarnak | last3 = Valette | first3 = Alain | doi = 10.1017/CBO9780511615825 | isbn = 0-521-82426-5 | mr = 1989434 | publisher = Cambridge University Press, Cambridge | series = London Mathematical Society Student Texts | title = Elementary number theory, group theory, and Ramanujan graphs |title-link=Elementary Number Theory, Group Theory and Ramanujan Graphs| volume = 55 | year = 2003}} These remarkable Ramanujan graphs also have large expansion coefficient.

The odd girth and even girth of a graph are the lengths of a shortest odd cycle and shortest even cycle respectively.

The {{visible anchor|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.{{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

}}.

The girth of an undirected graph can be computed by running a breadth-first search from each node, with complexity $O(nm)$ where $n$ is the number of vertices of the graph and $m$ is the number of edges.{{Cite web |url=http://webcourse.cs.technion.ac.il/234247/Winter2003-2004/ho/WCFiles/Girth.pdf |title=Question 3: Computing the girth of a graph |access-date=February 22, 2023 |archive-date=August 29, 2017 |archive-url=https://web.archive.org/web/20170829175217/http://webcourse.cs.technion.ac.il/234247/Winter2003-2004/ho/WCFiles/Girth.pdf |url-status=dead }} 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.{{Cite web |last=Völkel |first=Christoph Dürr, Louis Abraham and Finn |date=2016-11-06 |title=Shortest cycle |url=https://tryalgo.org/en/cycles/2016/11/06/shortest-cycle/ |access-date=2023-02-22 |website=TryAlgo |language=en-US}} Better algorithms are known in the case where the girth is even{{Cite web |title=ds.algorithms - Optimal algorithm for finding the girth of a sparse graph? |url=https://cstheory.stackexchange.com/questions/10983/optimal-algorithm-for-finding-the-girth-of-a-sparse-graph |access-date=2023-02-22 |website=Theoretical Computer Science Stack Exchange |language=en}} and when the graph is planar.{{Cite journal |last1=Chang |first1=Hsien-Chih |last2=Lu |first2=Hsueh-I. |date=2013 |title=Computing the Girth of a Planar Graph in Linear Time |journal=SIAM Journal on Computing |volume=42 |issue=3 |pages=1077–1094 |doi=10.1137/110832033 |arxiv=1104.4892 |s2cid=2493979 |issn=0097-5397}} 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.

Category:Graph invariants