How are graph theory and topology connected?

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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.

Graph theory and topology are deeply interconnected, sharing common historical roots, problems, and techniques. Both fields originated from the study of "position," with Leonhard Euler describing his work on the Königsberg bridge problem as geometria situs (geometry of position), which later evolved into the topological concept of analysis situs.1

The connection is further evidenced by Euler’s formula ( $V - E + F = 2$ ), which relates the vertices, edges, and faces of a polyhedron. By viewing the edges and vertices of a polyhedron as a graph embedded on a surface, mathematicians were able to generalize this formula to surfaces of different genera, such as a torus. This study of pasting polygons to construct surfaces laid the foundation for combinatorial and algebraic topology.1

This relationship fostered the subfield of topological graph theory, which examines:
* Planar graphs: Graphs that can be drawn on a plane or sphere without edges crossing.1
* Graph embedding: Determining which surfaces (like a torus) can host a graph without edge intersections.1
* Map-colouring problems: The study of coloring vertices or regions on various surfaces, such as the four-colour map problem.1