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Logistics Grid

A mining site consists of \(n\) Extraction Mines and \(m\) Refining Plants (where \(n, m \geq 1\)). For logistical efficiency, every mine must be connected to every plant by a direct road. In mathematics, we represent this as a complete bipartite graph, denoted by \(K_{n, m}\).

In this bipartite structure, the nodes are divided into two distinct sets: a set of \(n\) nodes representing the mines and a set of \(m\) nodes representing the plants. Connections only exist between nodes of different sets—never between two mines or two plants. To prevent traffic collisions, these roads are built on a single flat plain and must be planar (meaning no two roads are allowed to cross).

Problem Statement

Consider a "Feasibility Matrix" \(M\) for all \(n, m \geq 1\), where an entry \(M_{n,m} = 1\) if a planar layout for the graph \(K_{n, m}\) is possible, and \(0\) otherwise. Which of the following statements is true?

a) \(M_{n, m} = 0\) for all cases where \(n + m > 5\).

b) If you have only 2 mines (\(n = 2\)), you can connect them to any number of plants \(m\) without ever crossing a road, meaning \(M_{2, m} = 1\) for all \(m \geq 1\).

c) The graph \(K_{3,3}\) is planar because it is possible to draw it in a 2D plane without any roads intersecting.

d) Every graph \(K_{n, m}\) is planar as long as the number of mines (\(n\)) is different from the number of plants (\(m\)).

e) None of the above

Original idea by: Luis Alberto Vásquez Vargas

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