 CARMA DISCRETE MATHEMATICS SEMINAR
 Speaker: Dr Guillermo PinedaVillavicencio, Collaborative Research Network, The University of Ballarat
 Title: On the maximum order of graphs embedded in surfaces
 Location: Room V129, Mathematics Building (Callaghan Campus) The University of Newcastle
 Time and Date: 3:00 pm, Thu, 5^{th} Sep 2013
 Abstract:
Joint work with David Wood (Monash University, Australia) and Eran Nevo (BenGurion University of the Negev, Israel).
The maximum number of vertices of a graph of maximum degree $\Delta\ge 3$ and diameter $k\ge 2$ is upper bounded by $\Delta^{k}$. 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 $2\Delta^{\lfloor k/2\rfloor}$. The main result of this paper is that, for large $\Delta$, graphs embedded in surfaces of bounded Euler genus $g$ behave like trees. Specifically, we show that, for large $\Delta$, such graphs have orders bounded from above by
\begin{cases} (c_0g+c_1)\Delta^{\lfloor k/2\rfloor} & \text{if $k$ is even}\\
(c_0g^2+c_1)\Delta^{\lfloor k/2\rfloor} & \text{if $k$ is odd}
\end{cases}
where $c_0,c_1$ are absolute constants.
With respect to lower bounds, we construct graphs of Euler genus $g$, odd diameter and orders $(c_0\sqrt{g}+c_1)\Delta^{\lfloor k/2\rfloor}$, for absolute constants $c_0,c_1$.
Our results answer in the negative a conjecture by Miller and Širáň (2005). Before this paper, there were constructions of graphs of Euler genus $g$ and orders $c_0\Delta^{\lfloor k/2\rfloor}$ for an absolute constant $c_0$. Also, Šiagiová and Simanjuntak (2004) provided an upper bound of $(c_0g+c_1)k\Delta^{\lfloor k/2\rfloor}$ with absolute constants $c_0,c_1$.
 [Permanent link]
