The number of cycles in 2-factors of cubic graphs
Let \({\mathcal G}\) denote the family of all cubic 3-connected simple graphs, and let \({\mathcal G}_ p\) be the planar graphs in \({\mathcal G}\). Each graph in \({\mathcal G}\) has a 2-factor (by Petersen's Theorem). The object is to determine the number of (even) cycles needed in such 2-factors, so let \(k_ 2(G)\) \((k^ e_ 2(G))\) denote the smallest number of vertex disjoint (even) cycles that cover the vertices of G, and let \(f_ 0=\liminf | G_ n| /k_ 2(G_ p)\) for \(G_ n\in {\mathcal G}\), \(f_ 1=\liminf | G_ n| /k_ 2(G_ n)\) for \(G_ n\in {\mathcal G}_ p\), and \(f_ 2=\liminf | G_ n| /k^ e_ 2(G_ n)\) for \(G_ n\in {\mathcal G}_ p\). It is shown that \(f_ 0\leq 10\), \(f_ 1\leq 28\), and \(f_ 2\leq 31\), and also tht \(f_ 1\geq 6\).
scientific article; zbMATH DE number 4202308