Light structures in 1-planar graphs with an application to linear 2-arboricity

The linear 2-arboricity la(2) (G) of a graph G is the least integer k such that G can be partitioned into k edge-disjoint forests, whose component trees are paths of length at most 2. A graph is called 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. In this paper, we show that every 1-planar graph G of minimum degree at least 2 contains either a 29-light-edge or a 2-alternating-cycle. This result is applied to show that if the maximum degree of a 1-planar graph G is Delta, then la(2)(G) <= inverted left perpendicular Delta+1/2 inverted right perpendicular +14 Since la(2 )(G) >= inverted left perpendicular Delta/2 inverted right perpendicular for any graph G, our solution is within 14 from optimal. (C) 2019 Elsevier B.V. All rights reserved.