Gr\"{u}nbaum colorings of locally planar Fisk triangulations

Kenta Noguchi
Keio University

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Minisymposium: GRAPH IMBEDDINGS AND MAP SYMMETRIES

Content: A Gr\"{u}nbaum coloring of a triangulation $T$ of a surface $F^2$ is a 3-edge-coloring of $T$ such that every face of $T$ receives all the three colors. A Gr\"{u}nbaum coloring of $T$ 1-to-1 corresponds to a proper 3-edge-coloring of the dual $T^*$ of $T$. Robertson conjectured that every locally planar triangulation $T$ of a non-spherical surface has a Gr\"{u}nbaum coloring. Thomassen showed that every locally planar triangulation is 5-colorable. It is not difficult to see that there exists a Gr\"{u}nbaum coloring of $T$ if $T$ is 4-colorable. The only two families are known as 5-chromatic locally planar triangulations; One is a subset of Eulerian triangulations and the other is a set of Fisk triangulations. It is known that every locally planar Eulerian triangulation has a Gr\"{u}nbaum coloring. In this talk, we show that every locally planar Fisk triangulation has a Gr\"{u}nbaum coloring.

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