MINIMAX DEGREES OF QUASIPLANAR GRAPHS WITH NO SHORT CYCLES OTHER THAN TRIANGLES

Oleg V. Borodin, Anna O. Ivanova, Alexandr V. Kostochka, Naeem N. Sheikh
2008 Taiwanese journal of mathematics  
For an edge xy, let M (xy) be the maximum of the degrees of x and y. The minimax degree (or M -degree) of a graph G is M * (G) = min{M (xy)|xy ∈ E(G)}. In order to get upper bounds on the game chromatic number of planar graphs, He, Hou, Lih, Shao, Wang, and Zhu showed that every planar graph G without leaves and 4-cycles has minimax degree at most 8, which was improved by Borodin, Kostochka, Sheikh, and Yu to the sharp bound 7. We show that every planar graph G without leaves and 4-and 5-cycles
more » ... has M -degree at most 5, which bound is sharp. We also show that every planar graph G without leaves and cycles of length from 4 to 7 has M -degree at most 4, which bound is attained even on planar graphs with no cycles of length from 4 to arbitrarily large number. Besides, we give sufficient conditions for a planar graph to have M -degrees 3 and 2. Similar results are obtained for graphs embeddable into the projective plane, the torus and the Klein bottle. Dedicated to Professor Ko-Wei Lih on the occasion of his 60th birthday
doi:10.11650/twjm/1500404982 fatcat:t74n574o6rgu7ojhecmrrgusrq