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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
##### Keywords:
planar graphs; decomposition; degree; game coloring
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##### References:
 [1] Borodin, A generalization of Kotzig’s theorem and prescribed edge coloring of plane graphs, Math Notes Acad Sci USSR 48 pp 1186– (1990) [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 [4] Kotzig, Contribution to the theory of Eulerian polyhedra, Mat Čas 13 pp 20– (1963)
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