ON THE BAR VISIBILITY NUMBER OF COMPLETE BIPARTITE GRAPHS

A t-bar visibility representation of a graph assigns each vertex up to t horizontal bars in the plane so that two vertices are adjacent if and only if some bar for one vertex can see some bar for the other via an unobstructed vertical channel of positive width. The least t such that G has a t-bar visibility representation is the bar visibility number of G, denoted by b(G). For the complete bipartite graph K-m,K-n, the lower bound b(K-m,K-n) >= inverted right perpendicularmn+4/2m+2ninverted left perpendicular from Euler's formula is well known. We prove that equality holds.

Automated summary

The T-bar visibility representation assigns each vertex of a graph up to t horizontal bars in the plane, with two vertices being adjacent if one bar can see another bar through an unobstructed vertical channel. The bar visibility number of G, denoted by b(G), is the least t such that G has a t-bar visibility representation. The equality of the lower bound for the complete bipartite graph K-m,K-n from Euler's formula is proven to hold.