Abstract
A graph G has a (d,h)-decomposition if there is a pair (D,F) such that F is a subgraph of G and D is an acyclic orientation of G−E(F), 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 (3,1)-decomposable, and for {i,j}⊆{3,4,6}, toroidal graphs without i- and j-cycles are (2,1)-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 {i,j}⊆{3,4,6}.
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