\(M\)-degrees of quadrangle-free planar graphs.

*(English)*Zbl 1161.05024The \(M\)-degree of an edge \(xy\) in a graph \(G\) is the maximum of the degrees of \(x\) and \(y\); the \(M\)-degree of \(G\) is the minimum over the \(M\)-degrees of its edges W. He, X. Hou, K.-W. Lih, J. Shao, W. Wang and X. Zhu [J. Graph Theory 41, No. 4, 307–317 (2002; Zbl 1016.05033)] showed that every planar graph without leaves of 4-cycles has \(M\)-degree at most 8. The present authors find that the maximum \(M\)-degrees for planar and projective planar graphs with no leaves of 4-cycles is 7; for such graphs on the torus or Klein bottle, the maximum is 8. Both bounds are sharp.

Reviewer: Arthur T. White (Kalamazoo)

##### MSC:

05C10 | Planar graphs; geometric and topological aspects of graph theory |

05C35 | Extremal problems in graph theory |

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\textit{O. V. Borodin} et al., J. Graph Theory 60, No. 1, 80--85 (2009; Zbl 1161.05024)

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##### References:

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[2] | Borodin, Structural theorem on plane graphs with application to the entire coloring, J Graph Theory 23 pp 233– (1996) · Zbl 0863.05035 |

[3] | He, Edge-partitions of planar graphs and their game coloring numbers, J Graph Theory 41 pp 307– (2002) · Zbl 1016.05033 |

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