A (2, 1)-Decomposition of Planar Graphs Without Intersecting 3-Cycles and Adjacent 4--Cycles

Let G be a graph. For two positive integers d and h , a ( d , h )- decomposition of G is a pair ( G , H ) such that H is a subgraph of G of maximum degree at most h and D is an acyclic orientation of G - E ( H ) of maximum out-degree at most d . A graph G is ( d , h )- decomposable if G has a ( d , h )-decomposition. In this paper, we prove that every planar graph without intersecting 3-cycles and adjacent 4 - -cycles is (2, 1)-decomposable. As a corollary, we obtain that every planar graph without intersecting 3-cycles and adjacent 4 - -cycles has a matching M such that G - M is 2-degenerate and hence G - M is DP-3-colorable and Alon-Tarsi number of G - M is at most 3.