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arXiv · 1806.11333

Embeddings of 3-connected 3-regular planar graphs on surfaces of non-negative Euler characteristic

Kengo Enami
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Abstract

Whitney's theorem states that every 3-connected planar graph is uniquely embeddable on the sphere. On the other hand, it has many inequivalent embeddings on another surface. We shall characterize structures of a $3$-connected $3$-regular planar graph $G$ embedded on the projective-plane, the torus and the Klein bottle, and give a one-to-one correspondence between inequivalent embeddings of $G$ on each surface and some subgraphs of the dual of $G$ embedded on the sphere. These results enable us to give explicit bounds for the number of inequivalent embeddings of $G$ on each surface, and propose effective algorithms for enumerating and counting these embeddings.

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Kengo Enami. 2019-09-24. Embeddings of 3-connected 3-regular planar graphs on surfaces of non-negative Euler characteristic. https://doi.org/10.23638/dmtcs-21-4-14

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