Shortness parameters for polyhedral graphs (Q1304819)

For a family \({\mathcal F}\) of graphs the following shortness parameters are introduced: \[ \begin{aligned} \sigma ({\mathcal F}) & = \liminf_{G\in{\mathcal F}} \bigl\{\log h(G)/ \log n(G)\bigr\}, \\ \rho({\mathcal F}) & = \liminf_{G\in{\mathcal F}} \bigl\{h(G)/n(G)\},\\ \tau({\mathcal F}) & = \sup_{G\in {\mathcal F}}\bigl(n(G)-h(G)\bigr), \end{aligned} \] where \(n(G)\) is the order of \(G\) and \(h(G)\) is the circumference of \(G\). The author studies the above three parameter for various classes of polyhedral graphs. A number of known and new results are presented, and several conjectures are formulated.