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).
Bibliographical noteFunding Information:
Research of Nevo was partially supported by the Marie Curie grant IRG-270923 and the ISF grant 805/11 . Research of Pineda-Villavicencio was supported by a postdoctoral fellowship funded by the Skirball Foundation via the Center for Advanced Studies in Mathematics at Ben-Gurion University of the Negev, and by an ISF grant. Research of Wood is supported by the Australian Research Council .
© 2016 Elsevier Inc.
- Degree-diameter problem
- Graphs on surfaces
- Vertex separator