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.
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 G. Chartrand and L. Lesniak Graphs and Digraphs 3rd Edition (Chapman & Hall New York 1996).
 G.L. Chia and C.S. Gan On crossing numbers of 5-regular graphs in: J.-Y. Cai and C.K. Wong (Eds.) Computing and Combinatorics Lecture Notes in Comput. Sci. 2387 (2002) 230–237. doi:10.1007/3-540-45655-4_26
 G.L. Chia and C.S. Gan Minimal regular graphs with given girths and crossing numbers Discuss. Math. Graph Theory 24 (2004) 223–237. doi:10.7151/dmgt.1227
 Z.D. Ouyang J. Wang and Y.Q. Huang The crossing number of join of the generalized Petersen graph P(3 1) with path and cycle Discuss. Math. Graph Theory 38 (2018) 351–370. doi:10.7151/dmgt.2005
 M. Schaefer Crossing Numbers of Graphs (CRC Press Inc. Boca Raton Florida 2017).
 Y.S. Yang J.H. Lin and Y.J. Dai Largest planar graphs and largest maximal planar graphs of diameter two J. Comput. Appl. Math. 144 (2002) 349–358. doi:10.1016/S0377-0427(01)00572-6
SCImago Journal Rank (SJR) 2018: 0.763 Source Normalized Impact per Paper (SNIP) 2018: 0.934
Mathematical Citation Quotient (MCQ) 2018: 0.42
researchers in the fields of: colourings, partitions (general colourings), hereditary properties, independence and dominating structures (sets, paths, cycles, etc.), cycles, local properties, products of graphs