In the paper, crossing number of join product G*+Dn for disconnected graph G* consisting two components isomorphic to K2 and K3 is given, where Dn consists n isolated vertices. Presented proofs are completed with help configurations that a graphical representation minimum numbers crossings between different subgraphs whose edges do not cross G*. For first time, multiple symmetry presented as pa...

A k-stable c-coloured Candy Crush grid is a weak proper c-colouring of particular type k-uniform hypergraph. In this paper we introduce fully polynomial randomised approximation scheme (FPRAS) which counts the number grids given size (m,n) for certain values c and k. We implemented algorithm on Matlab, found that in with 7 available colours there are approximately 4.3×1061 3-stable colourings. ...