An Improvement to Chvátal and Thomassen’s Upper Bound for Oriented Diameter

Abstract

An orientation of an undirected graph G is an assignment of exactly one direction to each edge of G. The oriented diameter of a graph G is the smallest diameter among all the orientations of G. The maximum oriented diameter of a family of graphs $\mathscr {F}$ is the maximum oriented diameter among all the graphs in $\mathscr {F}$. Chvátal and Thomassen [JCTB, 1978] gave a lower bound of $\frac{1}{2}{d^2+d}$ and an upper bound of $2d^2+2d$ for the maximum oriented diameter of the family of 2-edge connected graphs of diameter d. We improve this upper bound to $1.373 d^2 + 6.971d-1$, which outperforms the former upper bound for all values of d greater than or equal to 8. For the family of 2-edge connected graphs of diameter 3, Kwok, Liu and West [JCTB, 2010] obtained improved lower and upper bounds of 9 and 11 respectively. For the family of 2-edge connected graphs of diameter 4, the bounds provided by Chvátal and Thomassen are 12 and 40 and no better bounds were known. By extending the method we used for diameter d graphs, along with an asymmetric extension of a technique used by Chvátal and Thomassen, we have improved this upper bound to 21.