  Partial cubes are graphs that can be isometrically embedded into hypercubes.
Convex cycles play an important role in the study of partial cubes. In this
paper, we prove that a regular partial cube is a hypercube (resp., a doubled
Odd graph, an even cycle of length $2n$ where $n\geqslant 4$) if and only if
all its convex cycles are 4-cycles (resp., 6-cycles, $2n$-cycles). In
particular, the partial cubes whose all convex cycles are 4-cycles are
equivalent to almost-median graphs, so we obtain that regular almost-median
graphs are exactly hypercubes, which generate the result by Mulder [J. Graph
Theory, 4 (1980) 107--110] -- regular median graphs are hypercubes.
