Abstract
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 I -graphs, a further generalization of generalized Petersen graphs. The characterization of Hamiltonian I -graphs follows from the fact that one can choose a 1 -factor in any I -graph 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 I -graphs as a subfamily.
Highlights
A graph is Hamiltonian if it contains a spanning cycle (Hamiltonian cycle)
Combining the definition of graph bundles with Theorem 3.3, we can find a family of connected cubicgraphs that contains the family of connected I-graphs as a subfamily
In a good Eulerian subgraph W there are two extreme cases: 1. each vertex of W is 4-valent: this means that W = X; in this case the complementary 2-factor Y = G − F is a Hamiltonian cycle and no edge of F is used; 2. each vertex of W is 2-valent: this means that W is a good Hamiltonian cycle in X
Summary
To find a Hamiltonian cycle in a graph is an NP–complete problem (see [12]) This fact implies that a characterization result for Hamiltonian graphs is hard to find. Using this condition we can completely characterize the Hamiltonian I-graphs. A characterization of Hamiltonian generalized Petersen graphs was obtained by Alspach [2]. A generalized Petersen graph G(n, q) is Hamiltonian if and only if it is not isomorphic to G(n, 2) when n ≡ 5 (mod 6). Combining the definition of graph bundles with Theorem 3.3, we can find a family of connected cubic (multi)graphs that contains the family of connected I-graphs as a subfamily (see Section 5).
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