Petersen's theorem states that every cubic graph with no bridges has a perfect matching (Petersen 1891; Frink 1926; König 1936; Skiena 1990, p. 244). In fact, this theorem can be extended to read, "every cubic graph with 0, 1, or 2 bridges has a perfect matching."
The graph above shows the smallest counterexample for 3 bridges, namely a connected cubic graph on 16 vertices having no perfect matchings. This graph is implemented in the Wolfram Language as GraphData["Cubic", 16, 14].
Errera (1922) strengthened Petersen's theorem by proving that if all bridges of a connected cubic graph lie on a single path of , then has a perfect matching.