An Improved Upper Bound on the Linear 2-arboricity of 1-planar Graphs

Abstract

The linear 2-arboricity la2(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. In this paper, we prove that if G is a 1-planar graph with maximum degree Δ, then $${\rm{l}}{{\rm{a}}_2}\left( G \right) \le \left\lceil {{{{\rm{\Delta + 1}}} \over 2}} \right\rceil + 7$$ . This improves a known result of Liu et al. (2019) that every 1-planar graph G has $${\rm{l}}{{\rm{a}}_2}\left( G \right) \le \left\lceil {{{{\rm{\Delta + 1}}} \over 2}} \right\rceil + 14$$ . We also observe that there exists a 7-regular 1-planar graph G such that $${\rm{l}}{{\rm{a}}_2}\left( G \right) = 6 = \left\lceil {{{{\rm{\Delta + 1}}} \over 2}} \right\rceil + 2$$ , which implies that our solution is within 6 from optimal.