Abstract
For each surface \(\Sigma \), we define \(\Delta (\Sigma ) =\) max\(\{\Delta (G)|\,G\) is a class two graph with maximum degree \(\Delta (G)\) that can be embedded in \(\Sigma \}\). Hence Vizing’s Planar Graph Conjecture can be restated as \(\Delta (\Sigma )=5\) if \(\Sigma \) is a sphere. For a surface \(\Sigma \), if \(\chi (\Sigma ) \in \{-5, -4, \dots , 0\}\), \(\Delta (\Sigma )\) is already known. In this paper, we show that \(\Delta (\Sigma )=10\) if \(\Sigma \) is a surface of characteristic \(\chi (\Sigma ) \in \{-6, -7\}\).
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Acknowledgements
R. Luo was supported by NSFC under Grant Numbers 11401003, and Z. Miao was supported by NSFC under Grant Numbers 11171288 and 11571149.
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Horacek, K., Luo, R., Miao, Z. et al. Finding \(\Delta (\Sigma )\) for a Surface \(\Sigma \) of Characteristic \(-6\) and \(-7\) . Graphs and Combinatorics 33, 929–944 (2017). https://doi.org/10.1007/s00373-017-1780-9
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DOI: https://doi.org/10.1007/s00373-017-1780-9