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From a Graph to the Plane

Source: Komal A. 683, based on a problem of the Miklós Schweitzer competition

October 6, 2021
functiongeometrygeometry unsolvedPlane Geometrygraph theorycombinatorics

Problem Statement

Let K=(V,E)K=(V, E) be a finite, simple, complete graph. Let ϕ:E→R2\phi: E \to \mathbb{R}^2 be a map from the edge set to the plane, such that the preimage of any point in the range defines a connected graph on the entire vertex set VV, and the points assigned to the edges of any triangle are collinear. Show that the range of ϕ\phi is contained in a line.
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