Linear 2-arboricity of planar graphs with 5-cycles not adjacent to short cycles

Abstract

Let G be a planar graph with 5-cycles not adjacent to t-cycles, i e {3,4}. We prove that if G is connected and 6(G) > 2, then G contains an edge xy with d(x)-f d(y) < 10 or a 2-alternating cycle. By this result, we obtain that its linear 2-arboricity/02(G) < \y[ +5. © 2019 Utilitas Mathematica Publishing Inc.. All rights reserved.