Editing Bridge (graph theory) (section)

==Bridgeless graphs==
A '''bridgeless graph''' is a graph that does not have any bridges. Equivalent conditions are that each [[Connected component (graph theory)|connected component]] of the graph has an [[ear decomposition|open ear decomposition]],<ref name="robbins39">{{citation
 | last = Robbins | first = H. E. | authorlink = Herbert Robbins
 | journal = [[The American Mathematical Monthly]]
 | pages = 281–283
 | title = A theorem on graphs, with an application to a problem of traffic control
 | volume = 46
 | year = 1939
 | issue = 5 | doi=10.2307/2303897| jstor = 2303897 | hdl = 10338.dmlcz/101517
 | hdl-access = free
 }}.</ref> that each connected component is [[K-edge-connected graph|2-edge-connected]], or (by [[Robbins' theorem]]) that every connected component has a [[strong orientation]].<ref name="robbins39"/>

An important open problem involving bridges is the [[cycle double cover conjecture]], due to [[Paul Seymour (mathematician)|Seymour]] and [[George Szekeres|Szekeres]] (1978 and 1979, independently), which states that every bridgeless graph admits a multi-set of simple cycles which contains each edge exactly twice.<ref>{{citation
 | last = Jaeger | first = F.
 | contribution = A survey of the cycle double cover conjecture
 | doi = 10.1016/S0304-0208(08)72993-1
 | pages = 1–12
 | series = North-Holland Mathematics Studies
 | title = Annals of Discrete Mathematics 27 – Cycles in Graphs
 | volume = 27
 | year = 1985| isbn = 978-0-444-87803-8
 }}.</ref>