Insights on the Middle-Layer Graph Frame

Abstract

Let k > 0 whenever n = 2k+1 for k E Z. While middle layer graphs such as M2,M3,...,Mk can be shortened to quotient graphs Mk/pi up to a fairly contracted frame, they can be shrunk even further so that they become into folding graphs R2,R3,...,Rk with worthwhile properties. Herein we will show, Rk vertices can be hued so that they are listed as hue-kinds H1,H2,....,Hn , n E N. Such kinds follow the catalan numbers C. Likewise we will show catalan numbers C1,....,Cn can be broken up to a ordered decomposition. Moreover, we will show they gather up to a rooted binary tree T'. In all, such framework helps to work out new ways to find hamilton cycles within the boolean mesh B. Indeed such findings give new inklings to further work.