A graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one another edge. In this work we prove that each 1-planar graph of minimum degree at least 3 contains an edge with degrees of its endvertices of type (3, ≤ 23) or (4, ≤ 11) or (5, ≤ 9) or (6, ≤ 8) or (7, 7). Moreover, the upper bounds 9, 8 and 7 here are sharp and the upper bounds 23 and 11 are very close to the possible sharp ones, which may be 20 and 10, respectively. This generalizes a result of Fabrici and Madaras [Discrete Math., 307 (2007) 854-865] which says that each 3-connected 1-planar graph contains a light edge, and improves a result of Hudák and Šugerek [Discuss. Math. Graph Theory, 32(3) (2012) 545-556], which states that each 1planar graph of minimum degree at least 4 contains an edge with degrees of its endvertices of type (4, ≤ 13) or (5, ≤ 9) or (6, ≤ 8) or (7, 7).

It is proved that the vertex set of any simple graph G can be equitably partitioned into k subsets for any integer k ≥ max{ ∆(G)+1 2 , |G| 4 } so that each of them induces a linear forest.

The cluster of a crossing in a graph drawing in the plane is the set of the four end-vertices of its two crossed edges. Two crossings are independent if their clusters do not intersect. In this paper, we prove that every plane graph with independent crossings has an equitable partition into m induced forests for any m ≥ 8. Moreover, we decrease this lower bound 8 for m to 6, 5, 4 and 3 if we additionally assume that the girth of the considering graph is at least 4, 5, 6 and 26, respectively.

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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 ∆(G) + 2p − 2 provided that ∆(G) ≥ 8p + 2 and p ≥ 2, and show that every 1-planar graph has an equitable edge coloring with k colors for any integer k ≥ 18. These three results respectively generalize the main theorems of three different previously published papers.