Geodesic
Light classes of generalized stars in polyhedral maps on surfaces
Discussiones Mathematicae. Graph Theory, Tome 24 (2004) no. 1, pp. 85-107

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A generalized s-star, s ≥ 1, is a tree with a root Z of degree s; all other vertices have degree ≤ 2. S_i denotes a generalized 3-star, all three maximal paths starting in Z have exactly i+1 vertices (including Z). Let be a surface of Euler characteristic χ() ≤ 0, and m():= ⎣(5 + √49-24χ( ))/2⎦. We prove:
Keywords: polyhedral maps, embeddings, light subgraphs, path, star, 2-dimensional manifolds, surface 
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Jendrol', Stanislav; Voss, Heinz-Jürgen. Light classes of generalized stars in polyhedral maps on surfaces. Discussiones Mathematicae. Graph Theory, Tome 24 (2004) no. 1, pp. 85-107. http://geodesic.mathdoc.fr/item/DMGT_2004_24_1_a7/