Geodesic
A Note on the Crossing Numbers of 5-Regular Graphs
Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 4, pp. 1127-1140

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The crossing number cr(G) of a graph G is the smallest number of edge crossings in any drawing of G. In this paper, we prove that there exists a unique 5-regular graph G on 10 vertices with cr(G) = 2. This answers a question by Chia and Gan in the negative. In addition, we also give a new proof of Chia and Gan’s result which states that if G is a non-planar 5-regular graph on 12 vertices, then cr(G) ≥ 2.
Keywords: crossing number, 5-regular graph, drawing 
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Ouyang, Zhangdong. A Note on the Crossing Numbers of 5-Regular Graphs. Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 4, pp. 1127-1140. http://geodesic.mathdoc.fr/item/DMGT_2020_40_4_a12/