M‐degrees of quadrangle‐free planar graphs
Oleg Veniaminovich Borodin, Alexandr V. Kostochka, Naeem Nisar Sheikh, Gexin Yu · Journal of Graph Theory · 2008
Abstract The M‐degree of an edge xy in a graph is the maximum of the degrees of x and y. The M‐degree of a graph G is the minimum over M‐degrees of its edges. In order to get upper bounds on the game chromatic number, He et al showed that every planar graph G without leaves and 4‐cycles has M‐degree at most 8 and gave an example of such a graph with M‐degree 3. This yields upper bounds on the game chromatic number of C4‐free planar graphs. We determine the maximum possible M‐degrees for planar, projective‐planar and toroidal graphs without leaves and 4‐cycles. In particular, for planar and projective‐planar graphs this maximum is 7. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 80–85, 2009