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. In this paper, we prove that the 3-cycle is light in the familyof 1-planar graphs with minimum vertex degree at least 5 and minimumedge degree at least 12. This generates a known result of Fabrici andMadaras [8]. 1. IntroductionAll graphs considered in the paper are nite, simple and undirected. Weuse V (G), E(G), F(G), δ(G) and ∆(G) to denote the set of vertices, the setof edges, the set of faces, the minimum degree and the maximum degree ofa plane graph G, respectively. The degree of an edge uv in G is the value ofd G (u) + d G (v). A k-, k + - and k − -vertex(resp.face) is a vertex (resp.face) ofdegree k, at least k and at most k, respectively. For other undened conceptswe refer the reader to [2].A graph is 1-planar if it can be drawn on a plane so that each edge is crossedby at most one other edge. The notion of 1-planarity was introduced by Ringel[14], who proved that each 1-planar graph is vertex 7-colorable. This is therst result on the colorings of 1-planar graphs, and from then on, many authorsstartedto investigatethe coloringproblems(see[1, 3, 4, 6, 15, 16, 17, 21, 23, 24])and the structural properties (see [5, 8, 9, 10, 11, 13, 18, 19, 20, 22]) of 1-planargraphs. One of possible approaches in the study of local graph structures canbe formalized in the following way (see [12]):Let H be a connected graph and G be a family of graphs. If for any graphG ∈ G, G contains a subgraph K ≃ H such that max

LIGHT 3-CYCLES IN 1-PLANAR GRAPHS WITH DEGREE RESTRICTIONS