There is a question from the book, Prove that σ(G) ≤ cr (G).
How to prove it?
Thanks
In general, the thickness of a graph G , denoted by θ( G ), is the minimum number of planar subgraphs in a decomposition of G into planar subgraphs. If G is graph with p vertices and q edges then q/(3p-6)≤ θ( G ). The splitting number σ ( G ) of a graph G is the smallest K such that G can be obtained from a planar graph by K vertex identifications. In other words, the splitting number is the minimum number of vertex splittings required to transform G into a planar graph.
In general, there are many ways to draw G in the plane with no multiple crossings. Of all these drawings, there will be at least one that has a minimum number of crossings. This minimum number is called the crossing number of G , denoted cr ( G ). Such a drawing with a minimum number of crossings is always a simple drawing.
