Pentagonal 3-polytopal graphs with edges of only two types and shortness parameters
For a graph \(G\), let \(h(G)\) denote the length of a maximum cycle of \(G\). For an infinite class of graphs, \(\mathcal G\), the shortness exponent \(\sigma({\mathcal G})\) and the shortness coefficient \(\rho({\mathcal G})\) are defined by \[ \sigma({\mathcal G})= \lim \inf_{G\in {\mathcal G}} {\log h(G)\over \log| V(G)|}\quad\text{and}\quad \rho({\mathcal G})= \lim \inf_{G\in {\mathcal G}} {h(G)\over | V(G)|}. \] This paper considers the class of pentagonal 3-polytopal graphs all of whose edges are incident either with two 3-valent vertices or with a 3-valent vertex and a \(q\)-valent vertex. For most values of \(q\), the authors show that the shortness exponent of the class and the shortness coefficient of a special class are less than one. For \(q= 4\), they find a positive lower bound for the shortness coefficient.
scientific article; zbMATH DE number 719464
scientific article