Geodesic
Cycles through specified vertices in triangle-free graphs
Discussiones Mathematicae. Graph Theory, Tome 27 (2007) no. 1, pp. 179-191

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Let G be a triangle-free graph with δ(G) ≥ 2 and σ₄(G) ≥ |V(G)| + 2. Let S ⊂ V(G) consist of less than σ₄/4+ 1 vertices. We prove the following. If all vertices of S have degree at least three, then there exists a cycle C containing S. Both the upper bound on |S| and the lower bound on σ₄ are best possible.
Keywords: cycle, path, triangle-free graph 
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Paulusma, Daniel; Yoshimoto, Kiyoshi. Cycles through specified vertices in triangle-free graphs. Discussiones Mathematicae. Graph Theory, Tome 27 (2007) no. 1, pp. 179-191. http://geodesic.mathdoc.fr/item/DMGT_2007_27_1_a15/