A 1-factorization F of a complete graph K2n is said to be G-regular, or regular under G, if G is an automorphism group of F acting sharply transitively on the vertex-set. The problem of determining which groups can realize such a situation dates back to a result by Hartman and Rosa (Eur J Comb 6:45–48, 1985) on cyclic groups and it is still open when n is even, although several classes of groups were tested in the recent past. It has been recently proved, see Rinaldi (Australas J Comb 80(2):178–196, 2021) and Mazzuoccolo et al. (Discret Math 342(4):1006–1016, 2019), that a G-regular 1-factorization, together with a complete set of rainbow spanning trees, exists for each group G of order 2n, n odd. The existence for each even n>2 was proved when either G is cyclic and n is not a power of 2, or when G is a dihedral group. Explicit constructions were given in all these cases. In this paper we extend this result and give explicit constructions when n>2 is even and G is either abelian but not cyclic, dicyclic, or a non cyclic 2-group with a cyclic subgroup of index 2.

Quaternionic 1-Factorizations and Complete Sets of Rainbow Spanning Trees / Rinaldi, Gloria. - In: GRAPHS AND COMBINATORICS. - ISSN 0911-0119. - 39:1(2023), pp. 1-25. [10.1007/s00373-023-02610-6]

Quaternionic 1-Factorizations and Complete Sets of Rainbow Spanning Trees

Gloria Rinaldi
2023

Abstract

A 1-factorization F of a complete graph K2n is said to be G-regular, or regular under G, if G is an automorphism group of F acting sharply transitively on the vertex-set. The problem of determining which groups can realize such a situation dates back to a result by Hartman and Rosa (Eur J Comb 6:45–48, 1985) on cyclic groups and it is still open when n is even, although several classes of groups were tested in the recent past. It has been recently proved, see Rinaldi (Australas J Comb 80(2):178–196, 2021) and Mazzuoccolo et al. (Discret Math 342(4):1006–1016, 2019), that a G-regular 1-factorization, together with a complete set of rainbow spanning trees, exists for each group G of order 2n, n odd. The existence for each even n>2 was proved when either G is cyclic and n is not a power of 2, or when G is a dihedral group. Explicit constructions were given in all these cases. In this paper we extend this result and give explicit constructions when n>2 is even and G is either abelian but not cyclic, dicyclic, or a non cyclic 2-group with a cyclic subgroup of index 2.
2023
27-gen-2023
39
1
1
25
Quaternionic 1-Factorizations and Complete Sets of Rainbow Spanning Trees / Rinaldi, Gloria. - In: GRAPHS AND COMBINATORICS. - ISSN 0911-0119. - 39:1(2023), pp. 1-25. [10.1007/s00373-023-02610-6]
Rinaldi, Gloria
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11380/1295975
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