On the optimal layout of balanced complete multipartite graphs into grids and tree related structures

Optimization of layout for embedding half hypercube into conventional tree architectures

“…Until then only approximations were given in the form of bounds 10,15,17 . Here are few works which are concerned about optimizing the layout for the considered embedding: complete Josephus cube into tree related architectures, 8 balanced complete multipartite graphs onto cartesian product between {Path, Cycle} and trees, 18 balanced complete multipartite graphs onto grids and tree related structures, 19 spined cube into grid, 20 complete bipartite graph into sibling tree, 21 augmented cube into tree related and windmill structures, 9 fault tolerance mapping of ternary $$$ N $$$‐cube onto chips, 4 hierarchical cube into linear array and $$$ k $$$‐rooted complete binary trees, 5 hierarchical folded cubes into linear arrays and complete binary trees, 1 circular layout of hypercube, 12 hypercube into certain trees, 6 locally twisted cube into grid, 13 familiar graphs onto hypercube, 7 hypercube into cylinder, 14 and 3‐Ary $$$ n $$$‐cube into grid 22 …”

“…With regard to hypercube and variants of hypercube almost every work solely relies on lexicographic ordering 8,9,13,16,20 . There are few exceptions like References 18 and 19 wherein the approach of using lexicographic ordering was not considered.…”