Linear 2-Arboricity of Planar Graphs with Neither 3-Cycles Nor Adjacent 4-Cycles

Abstract

Let G be a planar graph with neither 3-cycles nor adjacent 4-cycles. We prove that if G is connected and ?(G) ? 2, then G contains an edge uv with d(u) + d(v) ≤ 7 or a 2-alternating cycle. By this result, we obtain that G's linear 2-arboricity $${la_{2}(G)\leq\lceil\frac{\Delta(G)+1}{2}\rceil+4.}$$ .