Proof of a theorem of Tutte using permanents

Let G be a graph. A well-known theorem due to Tutte states that G contains { 1 , 2 } -factor if and only if for every subset S ⊆ V ( G ) , i ( G \ S ) ⩽ | S | , where i ( G \ S ) denotes the number of isolated vertices of G \ S . In this note we give a new proof for this result using permanents.