Planar graphs without 5-cycles or without 6-cycles

Let G be a planar graph without 5-cycles or without 6-cycles. In this paper, we prove that if G is connected and δ ( G ) ≥ 2 , then there exists an edge x y ∈ E ( G ) such that d ( x ) + d ( y ) ≤ 9 , or there is a 2-alternating cycle. By using the above result, we obtain that (1) its linear 2-arboricity l a 2 ( G ) ≤ ⌈ Δ ( G ) + 1 2 ⌉ + 6 , (2) its list total chromatic number is Δ ( G ) + 1 if Δ ( G ) ≥ 8 , and (3) its list edge chromatic number is Δ ( G ) if Δ ( G ) ≥ 8 .