Projects per year
Abstract
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))(δ1)^{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))(δ1)^{⌊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)(δ1)⌊k/2⌋if k is evenc(g3/2+1)(δ1)⌊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)(δ1)⌊k/2⌋ for some absolute constant c>0. Our results answer in the negative a question of Miller and Širáň (2005).
Original language  English 

Pages (fromto)  2841 
Number of pages  14 
Journal  Journal of Combinatorial Theory, Series B 
Volume  119 
DOIs  
Publication status  Published  1 Jul 2016 
Keywords
 Degreediameter problem
 Graphs on surfaces
 Surface
 Vertex separator
Projects
 1 Finished

Graph colouring via entropy compression
Australian Research Council (ARC)
2/01/14 → 31/12/17
Project: Research