Geodesic
Light Graphs In Planar Graphs Of Large Girth
Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 1, pp. 227-238

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A graph H is defined to be light in a graph family if there exist finite numbers ϕ (H, 𝒢 ) and w(H,𝒢 ) such that each G ∈𝒢 which contains H as a subgraph, also contains its isomorphic copy K with Δ_G (K) ≤ϕ (H, 𝒢 ) and Σ_ x ∈ V(K) deg_G (x) ≤ w(H, 𝒢). In this paper, we investigate light graphs in families of plane graphs of minimum degree 2 with prescribed girth and no adjacent 2-vertices, specifying several necessary conditions for their lightness and providing sharp bounds on ϕ and w for light K_1,3 and C_10.
Keywords: planar graph, girth, light graph 
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Hudák, Peter; Maceková, Mária; Madaras, Tomáš; Široczki, Pavol. Light Graphs In Planar Graphs Of Large Girth. Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 1, pp. 227-238. http://geodesic.mathdoc.fr/item/DMGT_2016_36_1_a15/