A Note on the Crossing Numbers of 5-Regular Graphs

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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.

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Target audience:

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

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