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Abstract
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In this paper, we prove that there is no r-regular graph (r ≥ 2) with a
unique perfect matching. Also we show that a 2r-regular graph of order
n has at least (r−(rk−+1) k)rr−−kk−1n 2k-factors, where k ≤ r. We investigate
graphs with a unique [a, b]-factor and among other results, we prove that
a connected graph with minimum degree at least 2 and a unique [1, 2]-
factor with regular components is an odd cycle.
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