Simona Bonvicini, Gloria Rinaldi
A $1$-factorization of the complete multigraph $λK_{2n}$ is said to be indecomposable if it cannot be represented as the union of $1$-factorizations of $λ_0 K_{2n}$ and $(λ-λ_0) K_{2n}$, where $λ_0<λ$. It is said to be simple if no $1$-factor is repeated. For every $n\geq 9$ and for every $(n-2)/3\leqλ\leq 2n$, we construct an indecomposable $1$-factorization of $λK_{2n}$ which is not simple. These $1$-factorizations provide simple and indecomposable $1$-factorizations of $λK_{2s}$ for every $s\geq 18$ and $2\leqλ\leq 2\lfloor s/2\rfloor-1$. We also give a generalization of a result by Colbourn et al. which provides a simple and indecomposable $1$-factorization of $λK_{2n}$, where $2n=p^m+1$, $λ=(p^m-1)/2$, $p$ prime.
Simona Bonvicini, Tomaž Pisanski, Arjana Žitnik
A bicirculant is a regular graph that admits an automorphism having two vertex-orbits of the same size. A bicirculant can be described as follows. Given an integer $m \ge 1$ and sets $R, S, T \subseteq \mathbb Z_m$ such that $R=-R$, $T=-T$, $0 \not\in R \cup T$ and $0 \in S$, the graph $B(m;R,S,T)$ has vertex set $V=\{u_0,\dots,u_{m-1},v_0,\dots,v_m-1\}$ and edge set $E=\{u_iu_{i+j}| \ i \in\mathbb Z_m, j \in R\} \cup \{v_iv_{i+j}| \ i \in\mathbb Z_m, j \in T\} \cup\{u_iv_{i+j}| \ i \in\mathbb Z_m, j \in S\}.$ Bicirculant graphs with $R=T=\emptyset$ are known as cyclic Haar graphs. In 2025 we conjectured that the only non-hamiltonian graphs among regular connected bicirculants of degree more than one are the generalized Petersen graphs $G(m,2)$ with $m \equiv 5 \pmod 6$. Recently we have verified the conjecture for bicirculants with $|S|\le 2$ and for bicirculants with $|R|=|T|$ odd. In this paper we show that the conjecture holds for all bicirculants with $|S| \le 3$ and for all bicirculants with $|S| \ge 4$ and $m/\gcd(m, S)$ even. As a byproduct of our results, we prove that every connected bicirculant graph on $2m$ vertices with $|S| \ge 4$ is hamiltonian for even $m< 9\, 240$, and for odd $m< 3\,465$. Finally, we show that the existence of a hamilton cycle in every connected cyclic Haar graph of valence at least $4$ implies that every connected bicirculant graph of valence at least $4$ is hamiltonian.
Simona Bonvicini, Marco Buratti
We give a sharply-vertex-transitive solution of each of the nine Hamilton-Waterloo problems left open by Danziger, Quattrocchi and Stevens.
Simona Bonvicini, Tomaž Pisanski
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.
S. Bonvicini, T. Pisanski, A. Žitnik
A bicirculant is a regular graph that admits a semi-regular automorphism with two vertex-orbits of the same size. By $m$ we denote the size of vertex-orbits and by $d$ the valence of a bicirculant. Furthermore, we denote by $s$ the valence of the bipartite graph joining the two vertex-orbits. In 1983, Brian Alspach proved that the only non-hamiltonian generalized Petersen graphs are $G(m,2)$ with $m \equiv 5 \pmod 6$. In a recent paper we conjectured that this is the only exception among regular, connected bicirculants of degree $d > 1$ and we have verified the conjecture for the quartic bicirculants with $s=2$, also known as the generalized rose window graphs. In this paper we develop tools and apply them for a partial verification of the conjecture. We show that the conjecture holds for all bicirculants with $s \leq 2$. As a consequence we obtain that every connected bicirculant with $s \ge 3$ is hamiltonian if $m$ is a product of at most three prime powers. In particular, every connected bicirculant with $s \ge 3$ is hamiltonian for even $m<210$ and odd $m < 1155$. Our results imply that many other families of bicirculants are hamiltonian. For example, all bicirculants with $d-s$ odd are hamiltonian.
Simona Bonvicini, Tomaž Pisanski, Arjana Žitnik
A bicirculant is a regular, $d$-valent graph that admits a semiregular automorphism of order $m$ having two vertex-orbits of size $m$. The vertices of each orbit induce a circulant graph of order $m$ and the remaining edges span a regular bipartite graph of valence, say $s$, $1 \leq s \leq d$, connecting the two vertex-orbits. Generalized Petersen graphs constitute a prominent family of bicirculants, with $d = 3$ and $s = 1$. In 1983, Brian Alspach proved that all generalized Petersen graphs are hamiltonian, except for the family $G(m, 2)$ with $m\equiv 5\pmod 6$. In this paper we conjecture that among all connected bicirculants of valence at least 2, there are no other exceptions. It follows from various sources that the conjecture is true for all cubic bicirculants. In this paper we prove the conjecture for quartic bicirulants with $s = 2$, also known as the generalized rose window graphs.
Simona Bonvicini, Giuseppe Mazzuoccolo
Let m be a positive integer and let G be a cubic graph of order 2n. We consider the problem of covering the edge-set of G with the minimum number of matchings of size m. This number is called excessive [m]-index of G in literature. The case m=n, that is a covering with perfect matchings, is known to be strictly related to an outstanding conjecture of Berge and Fulkerson. In this paper we study in some details the case m=n-1. We show how this parameter can be large for cubic graphs with low connectivity and we furnish some evidence that each cyclically 4-connected cubic graph of order 2n has excessive [n-1]-index at most 4. Finally, we discuss the relation between excessive [n-1]-index and some other graph parameters as oddness and circumference.
Simona Bonvicini, Marco Buratti, Martino Garonzi, Gloria Rinaldi, Tommaso Traetta
Kirkman triple systems (KTSs) are among the most popular combinatorial designs and their existence has been settled a long time ago. Yet, in comparison with Steiner triple systems, little is known about their automorphism groups. In particular, there is no known congruence class representing the orders of a KTS with a number of automorphisms at least close to the number of points. We fill this gap by proving that whenever $v \equiv 39$ (mod 72), or $v \equiv 4^e48 + 3$ (mod $4^e96$) and $e \geq 0$, there exists a KTS on $v$ points having at least $v-3$ automorphisms. This is only one of the consequences of a careful investigation on the KTSs with an automorphism group $G$ acting sharply transitively on all but three points. Our methods are all constructive and yield KTSs which in many cases inherit some of the automorphisms of $G$, thus increasing the total number of symmetries. To obtain these results it was necessary to introduce new types of difference families (the doubly disjoint ones) and difference matrices (the splittable ones) which we believe are interesting by themselves.