The Linear Arboricity of Planar Graphs without \(5\)-cycles and \(6\)-cycles

Abstract

Let \(G\) be a planar graph with maximum degree \(\Delta\). It’s proved that if \(\Delta \geq 5\) and \(G\) does not contain \(5\)-cycles and \(6\)-cycles, then \(la(G) = \lceil\frac{\Delta(G)}{2}\rceil\).