On the maximum order of graphs embedded in surfaces

The maximum number of vertices in a graph of maximum degree δ≥3 and fixed diameter k≥2 is upper bounded by (1+o(1))(δ-1)k. If we restrict our graphs to certain classes, better upper bounds are known. For instance, for the class of trees there is an upper bound of (2+o(1))(δ-1)⌊k/2⌋ for a fixed k. The main result of this paper is that graphs embedded in surfaces of bounded Euler genus g behave like trees, in the sense that, for large δ, such graphs have orders bounded from above by(c(g+1)(δ-1)⌊k/2⌋if k is evenc(g3/2+1)(δ-1)⌊k/2⌋if k is odd, where c is an absolute constant. This result represents a qualitative improvement over all previous results, even for planar graphs of odd diameter k. With respect to lower bounds, we construct graphs of Euler genus g, odd diameter k, and order c(g+1)(δ-1)⌊k/2⌋ for some absolute constant c>0. Our results answer in the negative a question of Miller and Širáň (2005).