  The graphene-graphite relationship in structural geometry is a basic
principle to predict novel two-dimensional (2D) materials. Here, we demonstrate
that this is not the case in binary metallic systems. We use the Bayesian
optimization framework combined with the density-functional theory approach to
determine the stable configuration of atomic species on a hexagonal plane. We
show that the optimized structure of 2D Cu-Au exhibits the hexagonal lattice of
a hexagonal ring of Cu atoms containing one Au atom, where the number of the Cu
atoms is larger than that of the Au atoms in the unit cell, which is difficult
to speculate from the atomic distribution of CuAu in the L1$_0$ structure. We
also show that 2D Cu-$X$ with $X=$ Be, Zn, and Pd have hexagonal or elongated
rings containing different atoms in the unit cell. Based on the binary
Lennard-Jones model, we propose that such structures can appear for
weakly-bonded systems located in between the phase-separated and
strongly-bonded systems with the interatomic interaction energy between
different species.
