The Decomposability of Toroidal Graphs without Adjacent Triangles or Short Cycles

A graph G has a (Formula presented.) -decomposition if there is a pair (Formula presented.) such that F is a subgraph of G and D is an acyclic orientation of (Formula presented.), where the maximum degree of F is no more than h and the maximum out-degree of D is no more than d. This paper proves that toroidal graphs having no adjacent triangles are (Formula presented.) -decomposable, and for (Formula presented.), toroidal graphs without i- and j-cycles are (Formula presented.) -decomposable. As consequences of these results, toroidal graphs without adjacent triangles are 1-defective DP-4-colorable, and toroidal graphs without i- and j-cycles are 1-defective DP-3-colorable for (Formula presented.).