Berge–Fulkerson conjecture for bridgeless cubic graphs
Let be a bridgeless cubic graph. Berge–Fulkerson conjecture. The graph has six perfect matchings such that each edge of is covered by exactly two of them. This longstanding conjecture, introduced in 1971, is open; it is a central conjecture about perfect matchings in cubic graphs and implies several weaker conjectures.
References
Primary source
Paulo Magalhães Júnior and Antonio Kelson Silva, “On some perfect matching conjectures in infinite, cubic, bridgeless graphs”, arXiv:2607.29511 (2026).
Additional references
18 papers in this index state this conjecture (2012–2026). The statement above is taken from the most recent of them; the others are arXiv:2607.06396, arXiv:2607.16356, arXiv:2504.19201, arXiv:2411.09806, arXiv:2410.04389, arXiv:2206.10975, arXiv:2012.03259, arXiv:1904.02661, arXiv:1811.08363, arXiv:1807.08138, arXiv:1804.09449, arXiv:1702.07156, and 5 more.
Progress summary
Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.
Solutions 0
No solutions have been posted yet.