Geodesic
Inequalities involving the genus of a graph and its thicknesses†
Glasgow mathematical journal, Tome 7 (1965) no. 1, pp. 19-21

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Let G be a graph with p points and q lines, and genus γ. The thicknesst(G) has been introduced as the minimum number of planar subgraphs whose union is G. This topological invariant of a graph has been studied by Battle, Harary and Kodama [1], Tutte [7], Beineke, Harary and Moon [3], and Beineke and Harary [2].It is natural to generalise this concept of the thickness of a graph to the union of graphs with a specified genus. We say that the n-thickness of G is the minimum number of subgraphs of genus at most n whose union is G. Denoting the n-thickness of G by tn, we write in particular t0 the 0-thickness, i.e., the thickness. 
Beineke, Lowell W.; Harary, Frank. Inequalities involving the genus of a graph and its thicknesses†. Glasgow mathematical journal, Tome 7 (1965) no. 1, pp. 19-21. doi: 10.1017/S2040618500035097
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