Edge-oblique polyhedral graphs.
Let \(G=(V,E,F)\) be a polyhedral graph (i.e. 3-connected plane graph) with vertex set \(V\), edge set \(E\) and face set \(F\). An edge \(e\) is defined to be of type \((a,b; m,n)\) if it is incident with an \(a\)-vertex and a \(b\)-vertex and with an \(m\)-face and an \(n\)-face. Here an \(x\)-vertex and an \(r\)-face denote a vertex of degree \(x\) and a face of size \(r\), respectively. A graph which contains no two edges of the same type is called edge-oblique and if it contains at most \(z\) edges of each type it is called \(z\)-edge-oblique. In this paper it is proved that the set of edge-oblique graphs is finite. Several properties of edge-oblique graphs are proved. For example, it is shown that if \(G\) is an oblique graph then its maximum degree \(\Delta(G)\) is at most \(30\) and it has less than \(880\) edges. Another nice result of the paper states that the set of \(z\)-edge-oblique graphs is finite too. Note that an analogous problem concerning face-oblique graphs is considered in \textit{M. Voigt} and \textit{H. Walther} [Discrete Math. 244, 473--478 (2002; Zbl 0995.05039)].
scientific article; zbMATH DE number 2055511