arXiv subjects

Lorenzo Mella

Publications and source records attributed to Lorenzo Mella.

9 recordsLinked to original sources

arXiv2025

On relative simple Heffter spaces

In this paper, we introduce the concept of a relative Heffter space which simultaneously generalizes those of relative Heffter arrays and Heffter spaces. Given a subgroup $J$ of an abelian group $G$, a relative Heffter space is a resolvable configuration whose points form a half-set of $G\setminus{J}$ and whose blocks are all zero-sum in $G$. Here we present two infinite families of relative Heffter spaces satisfying the additional condition of being simple. As a consequence, we get new results on globally simple relative Heffter arrays, on mutually orthogonal cycle decompositions and on biembeddings of cyclic cycle decompositions of the complete multipartite graph into an orientable surface.

math.CO↗

arXiv2024

Heffter arrays over partial loops

A Heffter array over an additive group $G$ is any partially filled array $A$ satisfying that: (1) each one of its rows and columns sum to zero in $G$, and (2) if $i\in G\setminus\{0\}$, then either $i$ or $-i$ appears exactly once in $A$. In this paper, this notion is naturally generalized to that of $\mathcal{B}$-Heffter array over a partial loop, where $\mathcal{B}$ is a set of block-sum polynomials over an affine $1$-design on the set of entries in $A$.

math.CO↗

arXiv2024

The extended irregular domination problem

In this paper we introduce a new domination problem strongly related to the following one recently proposed by Broe, Chartrand and Zhang. One says that a vertex $v$ of a graph $Γ$ labeled with an integer $\ell$ dominates the vertices of $Γ$ having distance $\ell$ from $v$. An irregular dominating set of a given graph $Γ$ is a set $S$ of vertices of $Γ$, having distinct positive labels, whose elements dominate every vertex of $Γ$. Since it has been proven that no connected vertex transitive graph admits an irregular dominating set, here we introduce the concept of an extended irregular dominating set, where we admit that precisely one vertex, labeled with 0, dominates itself. Then we present existence or non existence results of an extended irregular dominating set $S$ for several classes of graphs, focusing in particular on the case in which $S$ is as small as possible. We also propose two conjectures.

math.CO↗

arXiv2023

A conjecture on the Crazy Knight's Tour Problem

Let $A$ be an $m\times n$ toroidal array containing filled and empty cells. Fix an orientation $R=(r_1,\dots,r_m)$ of each row and an orientation $C=(c_1,\dots,c_n)$ of each column of $A$. Given an initial filled cell $(i_1,j_1)$ consider the list $ L_{R,C}=((i_1,j_1),(i_2,j_2),\ldots,(i_k,j_k),$ $(i_{k+1},j_{k+1}),\ldots)$ where $j_{k+1}$ is the column index of the filled cell $(i_k,j_{k+1})$ of the row $R_{i_k}$ next to $(i_k,j_k)$ in the orientation $r_{i_k}$, and where $i_{k+1}$ is the row index of the filled cell of the column $C_{j_{k+1}}$ next to $(i_k,j_{k+1})$ in the orientation $c_{j_{k+1}}$. The problem is the following. Crazy Knight's Tour Problem: Do there exist $R$ and $C$ such that the list $L_{R,C}$ covers all the filled cells of $A$? This problem was introduced by Costa, Dalai and Pasotti to construct new biembeddings of graphs on surfaces starting from an Heffter array. Here we provide solution to the Crazy Knight's Tour Problem for infinite classes of cyclically $k$-diagonal square arrays, namely square arrays whose filled cells are exactly those of $k$ consecutive diagonals. These new constructions together with some known results induce us to propose the following. Conjecture: Let $A$ be a cyclically $k$-diagonal square array of order $n$. Then there exists a solution to the Crazy Knight's Tour Problem on $A$ if and only if $n$ and $k$ are odd integers with $n\geq k \geq3$.

math.CO↗

arXiv2023

A solution to the 7-prince's tour problem

In [1] the authors studied the closed tour problem on the $8\times 8$ chessboard of a chess piece, called $k$-prince, leaving open the existence of such a tour when $k=7$. In this note we find a solution to this open case.

math.GM↗

arXiv2023

Constructing generalized Heffter arrays via near alternating sign matrices

Let $S$ be a subset of a group $G$ (not necessarily abelian) such that $S\,\cap -S$ is empty or contains only elements of order $2$, and let $\mathbf{h}=(h_1,\ldots, h_m)\in \mathbb{N}^m$ and $\mathbf{k}=(k_1, \ldots, k_n)\in \mathbb{N}^n$. A generalized Heffter array GHA$^{\lambda}_S(m, n; \mathbf{h}, \mathbf{k})$ over $G$ is an $m\times n$ matrix $A=(a_{ij})$ such that: the $i$-th row (resp. $j$-th column) of $A$ contains exactly $h_i$ (resp. $k_j$) nonzero elements, and the list $\{a_{ij}, -a_{ij}\mid a_{ij}\neq 0\}$ equals $\lambda$ times the set $S\,\cup\, -S$. We speak of a zero sum (resp. nonzero sum) GHA if each row and each column of $A$ sums to zero (resp. a nonzero element), with respect to some ordering. In this paper, we use near alternating sign matrices to build both zero and nonzero sum GHAs, over cyclic groups, having the further strong property of being simple. In particular, we construct zero sum and simple GHAs whose row and column weights are congruent to $0$ modulo $4$. This result also provides the first infinite family of simple (classic) Heffter arrays to be rectangular ($m\neq n$) and with less than $n$ nonzero entries in each row. Furthermore, we build nonzero sum GHA$^{\lambda}_S(m, n; \mathbf{h}, \mathbf{k})$ over an arbitrary group $G$ whenever $S$ contains enough noninvolutions, thus extending previous nonconstructive results where $\pm S = G\setminus H$ for some subgroup $H$~of~$G$. Finally, we describe how GHAs can be used to build orthogonal decompositions and biembeddings of Cayley graphs (over groups not necessarily abelian) onto orientable surfaces.

math.CO↗

arXiv2023

A class of highly symmetric Archdeacon embeddings

Archdeacon, in his seminal paper $[1]$, defined the concept of Heffter array to provide explicit constructions of biembeddings of the complete graph $K_v$ into orientable surfaces, the so-called Archdeacon embeddings, and proved that these embeddings are $\mathbb{Z}_{v}$-regular. In this paper, we show that an Archdeacon embedding may admit an automorphism group that is strictly larger than $\mathbb{Z}_{v}$. Indeed, as an application of the interesting class of arrays recently introduced by Buratti in $[2]$, we exhibit, for infinitely many values of $v$, an embedding of this type having full automorphism group of size ${v \choose 2}$ that is the largest possible one.

math.CO↗

arXiv2023

Weak Heffter Arrays and biembedding graphs on non-orientable surfaces

In 2015, Archdeacon proposed the notion of Heffter arrays in view of its connection to several other combinatorial objects. In the same paper he also presented the following variant. A weak Heffter array $\mathrm{W}\mathrm{H}(m,n;h,k)$ is an $m \times n$ matrix $A$ such that: each row contains $h$ filled cells and each column contains $k$ filled cells; for every $x \in \mathbb{Z}_{2nk+1} \setminus \{0\}$, there is exactly one cell of $A$ whose element is one of the following: $x,-x,\pm x,\mp x$, where the upper sign on $\pm$ or $\mp$ is the row sign and the lower sign is the column sign; the elements in every row and column (with the corresponding sign) sum to $0$ in $\mathbb{Z}_{2nk+1}$. Also the ``weak concept'', as the classical one, is related to several other topics, such as difference families, cycle systems and biembeddings. Many papers on Heffter arrays have been published, while no one on weak Heffter arrays has been written. This is the first one and here we explore necessary conditions, existence and non-existence results, and connections to biembeddings into non-orientable surfaces.

math.CO↗

arXiv2022

Tight globally simple non-zero sum Heffter arrays and biembeddings

Square relative non-zero sum Heffter arrays, denoted by $\mathrm{N}\mathrm{H}_t(n;k)$, have been introduced as a variant of the classical concept of Heffter array. An $\mathrm{N}\mathrm{H}_t(n; k)$ is an $n\times n$ partially filled array with elements in $\mathbb{Z}_v$, where $v=2nk+t$, whose rows and whose columns contain $k$ filled cells, such that the sum of the elements in every row and column is different from $0$ (modulo $v$) and, for every $x\in \mathbb{Z}_v$ not belonging to the subgroup of order $t$, either $x$ or $-x$ appears in the array. In this paper we give direct constructions of square non-zero sum Heffter arrays with no empty cells, $\mathrm{N}\mathrm{H}_t(n;n)$, for every $n$ odd, when $t$ is a divisor of $n$ and when $t\in\{2,2n,n^2,2n^2\}$. The constructed arrays have also the very restrictive property of being "globally simple"; this allows us to get new orthogonal path decompositions and new biembeddings of complete multipartite graphs.

math.CO↗