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Abstract
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A graph is called equimatchable if all of its maximal matchings
have the same size. Kawarabayashi, Plummer, and Saito showed that the
only connected equimatchable 3-regular graphs are K4 and K3,3. We extend
this result by showing that for an odd positive integer r, if G is a connected
equimatchable r-regular graph, then G ∈ {Kr+1, Kr,r}. Also it is proved that
for an even r, a connected triangle-free equimatchable r-regular graph is
isomorphic to one of the graphs C5, C7, and Kr,r.
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