Simona Bonvicini

Octahedral, dicyclic and special linear solutions of some Hamilton-Waterloo problems

We give a sharply-vertex-transitive solution of each of the nine Hamilton-Waterloo problems left open by Danziger, Quattrocchi and Stevens.

Frattini-based starters in 2-groups

Let G be a group of order 2^t, with t>=4. We prove a sufficient condition for the existence of a one-factorization F of a complete graph, admitting G as an automorphism group acting sharply transitively on the vertex-set.

Hamiltonian cycles in I–graphs

Abstract The I–graphs generalize the family of generalized Petersen graphs. We show that a connected I–graph which is not a generalized Petersen graph is Hamiltonian.

On generalized null polarities

Assuming that a linear complex of planes without singular lines exists, the properties of the related generalized polarity are investigated.

Edge-Colorings of 4-Regular Graphs with the Minimum Number of Palettes

A proper edge-coloring of a graph G is an assignment of colors to the edges of G such that adjacent edges receive distinct colors. A proper edge-coloring defines at each vertex the set of colors of its incident edges. Following the terminology introduced by Horňák, Kalinowski, Meszka and Woźniak, we call such a set of colors the palette of the vertex. What is the minimum number of distinct palettes taken over all proper edge-colorings of G? A complete answer is known for complete graphs and cubic graphs. We study in some detail the problem for 4-regular graphs. In particular, we show that certain values of the palette index imply the existence of an even cycle decomposition of size 3 (a partition of the edge-set of a graph into 3 2-regular subgraphs whose connected components are cycles of even length). This result can be extended to 4d-regular graphs. Moreover, in studying the palette index of a 4-regular graph, the following problem arises: does there exist a 4-regular graph whose even cycle decompositions cannot have size smaller than 4?

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

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.

Abelian 1-factorizations in infinite graphs

For each finitely generated abelian infinite group G, we construct a 1-factorization of the countable complete graph admitting G as an automorphism group acting sharply transitively on vertices.