Colorings Of Plane Graphs With No Rainbow Faces

We prove that each n-vertex plane graph with girth g≥4 admits a vertex coloring with at least ⌈n/2⌉+1 colors with no rainbow face, i.e., a face in which all vertices receive distinct colors. This proves a conjecture of Ramamurthi and West. Moreover, we prove for plane graph with girth g≥5 that there is a vertex coloring with at least \( {\left\lceil {\frac{{g - 3}} {{g - 2}}n - \frac{{g - 7}} {{2{\left( {g - 2} \right)}}}} \right\rceil } \) if g is odd and \( {\left\lceil {\frac{{g - 3}} {{g - 2}}n - \frac{{g - 6}} {{2{\left( {g - 2} \right)}}}} \right\rceil } \) if g is even. The bounds are tight for all pairs of n and g with g≥4 and n≥5g/2−3.

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Correspondence to Veselin Jungić.

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* Supported in part by the Ministry of Science and Technology of Slovenia, Research Project Z1-3129 and by a postdoctoral fellowship of PIMS.

** Institute for Theoretical Computer Science is supported by Ministry of Education of CzechR epublic as project LN00A056.

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Jungić, V., Král’, D. & Škrekovski, R. Colorings Of Plane Graphs With No Rainbow Faces. Combinatorica 26, 169–182 (2006). https://doi.org/10.1007/s00493-006-0012-3

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Mathematics Subject Classification (2000):

  • 05C15