  We are interested in finding combinatorial criteria for embedding graphs into
torus and using them in the embedding check algorithm. It is a well-known fact
that any connected graph can be reduced to a one-vertex graph with loops and is
homotopy equivalent to such a graph. If a connected graph has intersection of
some edges, then some loops of one-vertex graph will also intersect. Therefore
the problem of embedding graphs into some surface is equivalent to the question
of embedding one-vertex graph in given surface. This paper considers a graph
with rotations or rotation system called hieroglyph. The equivalence of four
conditions is proved: (1) embedding hieroglyph into torus; (2) the absence of
forbidden subgraphs in a hieroglyph; (3) some condition for the graph of loops;
(4) the existence of a reduction of hieroglyph to one of the list of
hieroglyphs. We also prove the existence of an algorithm with complexity $O(n)$
that recognizes embedding hieroglyph into torus.
