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A 4-regular planar graph $G$ is said to be circle representable if thereexists a collection of circles drawn on the plane such that the touching andcrossing points correspond to the vertices of $G$, and the circular arcsbetween those points correspond to the edges of $G$. Lov\'asz (1970)conjectured that every 4-regular planar graph has a circle representation, butan infinite family of counterexamples was given by Bekos and Raftopoulou(2015). We reduce the order of the smallest known counterexamples among simplegraphs from 822 to 68 based on a multigraph counterexample of order 12.

Small 4-regular planar graphs that are not circle representable