A novel characterization of cubic Hamiltonian graphs via the associated quartic graphs

作者：Bonvicini Simona^*; Pisanski Tomaz

来源：Ars Mathematica Contemporanea, 2017, 12(1): 1-24.

• Generalized Petersen graphs
• I-graphs
• Hamiltonian cycles
• Eulerian tours
• Cayley multigraphs

We give a necessary and sufficient condition for a cubic graph to be Hamiltonian by analyzing Eulerian tours in certain spanning subgraphs of the quartic graph associated with the cubic graph by 1-factor contraction. This correspondence is most useful in the case when it induces a blue and red 2-factorization of the associated quartic graph. We use this condition to characterize the Hamiltonian Igraphs, a further generalization of generalized Petersen graphs. The characterization of Hamiltonian Igraphs follows from the fact that one can choose a 1-factor in any Igraph in such a way that the corresponding associated quartic graph is a graph bundle having a cycle graph as base graph and a fiber and the fundamental factorization of graph bundles playing the role of blue and red factorization. The techniques that we develop allow us to represent Cayley multigraphs of degree 4, that are associated to abelian groups, as graph bundles. Moreover, we can find a family of connected cubic (multi) graphs that contains the family of connected Igraphs as a subfamily.

• 出版日期2017