# On the bar visibility number of complete bipartite graphs

## TL;DR

This paper determines the exact bar visibility number for complete bipartite graphs, showing it equals a known lower bound derived from Euler's Formula, thus resolving a mathematical characterization.

## Contribution

It proves that the known lower bound for the bar visibility number of complete bipartite graphs is tight, establishing the exact value.

## Key findings

- The bar visibility number for $K_{m,n}$ equals the lower bound from Euler's Formula.
- The result provides a precise characterization of visibility representations for complete bipartite graphs.

## Abstract

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 $K_{m,n}$, the lower bound $b(K_{m,n})\ge\lceil{\frac{mn+4}{2m+2n}}\rceil$ from Euler's Formula is well known. We prove that equality holds.