Why can the complete graph K5 not be drawn on a plane without crossing edges?

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topology Because both a doughnut and a coffee cup have one hole (handle), they can be mathematically, or topologically, transformed into one another without cutting them in any way. For this reason, it has often been joked that topologists cannot tell the difference between a coffee cup and a doughnut.

In the field of graph theory, the complete graph $K_{5}$ is classified as a nonplanar graph. By definition, planar graphs are those that can be drawn on a plane or a sphere without any edges crossing except at their shared vertices.

$K_{5}$ consists of five vertices where every vertex is connected to every other vertex. Because of this high density of connections, it is mathematically impossible to embed the graph on a two-dimensional plane without at least one pair of edges intersecting between vertices. This property is a fundamental result in topological graph theory, which further establishes that any graph containing a structure equivalent to $K_{5}$ (or the bipartite graph $K_{3, 3}$ ) is inherently nonplanar. While $K_{5}$ cannot be drawn on a plane without crossings, it can be successfully embedded on surfaces of higher genus, such as a torus.1