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\(M\)-degrees of quadrangle-free planar graphs. (English) Zbl 1161.05024
The \(M\)-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.

MSC:
05C10 Planar graphs; geometric and topological aspects of graph theory
05C35 Extremal problems in graph theory
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References:
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