LAUSR

A graph is 1-planar if it can be drawn on a plane so that each edge is crossed by at most one other edge. In this paper, we first give a useful structural theorem for 1-planar graphs, and then apply it to the list edge and list total coloring, the (p, 1)-total labelling, and the equitable edge coloring of 1-planar graphs. More precisely, we verify the well-known List Edge Coloring Conjecture and List Total Coloring Conjecture for 1-planar graph with maximum degree at least 18, prove that the (p, 1)-total labelling number of every 1-planar graph G is at most $$\Delta (G)+2p-2$$Δ(G)+2p-2 provided that $$\Delta (G)\ge 8p+2$$Δ(G)≥8p+2 and $$p\ge 2$$p≥2, and show that every 1-planar graph has an equitable edge coloring with k colors for any integer $$k\ge 18$$k≥18. These three results respectively generalize the main theorems of three different previously published papers.