I have to prove that the Petersen graph has only one perfect match, I have one perfect match but I dont know how to prove formaly that it's unique.
2026-03-26 21:08:06.1774559286
Perfect matching in the Petersen graph
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The Petersen graph has six perfect matchings, not just one.
If you only found one, you probably found the one that's rotationally symmetric in the standard picture of the Petersen graph, with $5$-fold symmetry. There are five more (that are all rotations of the same pattern).
You can also look for matchings in the drawing of the Petersen graph with $3$-fold symmetry:
Here, the six perfect matchings come in two families of three (where each family consists of three rotations of the same matching). Because it's impossible to have a rotationally symmetric perfect matching in this drawing (the central vertex can only be matched to one of the outside vertices) it's easy to see that the number of perfect matchings must be a multiple of $3$.