The maximum number of vertices in a graph of maximum degree Ξβ₯3 and
fixed diameter kβ₯2 is upper bounded by (1+o(1))(Ξβ1)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))(Ξβ1)βk/2β 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 Ξ, such graphs have
orders bounded from above by begin{cases} c(g+1)(\Delta-1)^{\lfloor
k/2\rfloor} & \text{if $k$ is even} c(g^{3/2}+1)(\Delta-1)^{\lfloor k/2\rfloor}
& \text{if $k$ is odd}, \{cases} 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(gβ+1)(Ξβ1)βk/2β for some absolute constant
c>0. Our results answer in the negative a question of Miller and
\v{S}ir\'a\v{n} (2005).Comment: 13 pages, 3 figure