  For simply-laced Kac-Moody algebras $\frak g$, Stembridge (2003) proposed a
`local' axiomatization of crystal graphs of representations of $U_q(\frak g)$.
In this paper we propose axioms for edge-2-colored graphs which characterize
the crystals of integrable representations of $U_q(sp(4))$, regular crystal
graphs of $B_2$-type. An edge-colored directed graph which obeys our Axioms
(K0)--(K5) is called an R-{\em graph} (for brevity), and our main result is
that the regular crystals of $B_2$-type are R-graphs and vice versa. We give a
direct combinatorial construction for the crystals in question. On this way we
introduce a new, so-called {\em crossing model}, which does not exploit Young
tableaux. This combinatorial model consists of a two-component graph of a
rather simple form and of a certain set of integer-valued functions on its
vertices.
