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We are interested in finding combinatorial criteria for embedding graphs intotorus and using them in the embedding check algorithm. It is a well-known factthat any connected graph can be reduced to a one-vertex graph with loops and ishomotopy equivalent to such a graph. If a connected graph has intersection ofsome edges, then some loops of one-vertex graph will also intersect. Thereforethe problem of embedding graphs into some surface is equivalent to the questionof embedding one-vertex graph in given surface. This paper considers a graphwith rotations or rotation system called hieroglyph. The equivalence of fourconditions is proved: (1) embedding hieroglyph into torus; (2) the absence offorbidden 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 ofhieroglyphs. We also prove the existence of an algorithm with complexity $O(n)$that recognizes embedding hieroglyph into torus.

Embeddable of one-vertex graph with rotations into torus