On maximal graphical partitions

A partition of an integer $m$ is a sequence of nonnegative integers in nonincreasing order whose sum is equal to $m$. The length of a partition is the number of its nonzero parts. The set of all graphical partitions of $2m$, for a given $m$, is an order ideal of the lattice of all partitions of $2m$. We find new characterization of maximal graphical partitions and the number of maximal graphical partitions of length $n$. For each graphical partition $\lambda$ of integer $2m$ we construct maximal graphical partition $\mu$ of integer $2m$ with the same rank, which is dominate $\lambda$; also we find an algorithm that builds a sequence of elementary transformations from $\mu$ to $\lambda$.