arXiv subjects

Anita Pasotti

Publications and source records attributed to Anita Pasotti.

At least 19 recordsLinked to original sources

arXiv2026

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↗

arXiv2025

Definability of some $k$-ary Relations Over Second Order kinds of Logics

We consider the exprissibility in monadic second order logic of certain relations of importance in computer science. For integers $n\geq 1$ and $k\leq b$, a $k$-tuple of sequences in $\{0,1,\ldots, b-1\}^n$ are said to be $k$-hashed if there is a coordinate where they all differ. A set $\mathcal{C}$ of sequences is said to be a $k$-hash code if any $k$ distinct elements are $k$-hashed. Testing whether a code is $k$-hashing and determining the largest size of $k$-hash codes is an important problem in computer science. The use of general purpose solvers for this problem leads to question what minimal logic is needed to represent the problem. In this paper, we prove that the $k$-hashing relation on $k$-tuples is not definable in Monadic Second Order Logic (MSO), highlighting its limitations for this problem. Instead, the property can be expressed in extensions of the MSO that add the equi-cardinality relation.

math.LO↗

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

Shiftable Heffter spaces

The shiftable Heffter arrays are naturally generalized to the shiftable Heffter spaces. We present a recursive construction which starting from a single shiftable Heffter space leads to infinitely many other shiftable Heffter spaces of the same degree. We also present a direct construction making use of pandiagonal magic squares leading to a shiftable $(16\ell^2,4l;3)$ Heffter space for any $\ell \geq 1$. Combining these constructions we obtain a shiftable $(16\ell^2mn, 4\ell n; 3)$ Heffter space for every triple of positive integers $(\ell,m,n)$ with $m \geq n$.

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↗

arXiv2024

More Heffter Spaces via finite fields

A $(v,k;r)$ Heffter space is a resolvable $(v_r,b_k)$ configuration whose points form a half-set of an abelian group $G$ and whose blocks are all zero-sum in $G$. It was recently proved that there are infinitely many orders $v$ for which, given any pair $(k,r)$ with $k\geq3$ odd, a $(v,k;r)$ Heffter space exists. This was obtained by imposing a point-regular automorphism group. Here we relax this request by asking for a point-semiregular automorphism group. In this way the above result is extended also to the case $k$ even.

math.CO↗

arXiv2024

Heffter Spaces

The notion of a Heffter array, which received much attention in the last decade, is equivalent to a pair of orthogonal Heffter systems. In this paper we study the existence problem of a set of $r$ mutually orthogonal Heffter systems for any $r$. Such a set is equivalent to a resolvable partial linear space of degree $r$ whose parallel classes are Heffter systems: this is a new combinatorial design that we call a Heffter space. We present a series of direct constructions of Heffter spaces with block size odd and arbitrarily large degree $r$ obtained with the crucial use of finite fields. Among the applications we establish, in particular, the existence of $r$ mutually orthogonal $k$-cycle systems of order a prime power $q=2kw+1$ whenever $kw$ is odd and $w>4k^4\lceil{r\over k}\rceil$.

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↗

arXiv2022

Non-zero sum Heffter arrays and their applications

In this paper we introduce a new class of partially filled arrays that, as Heffter arrays, are related to difference families, graph decompositions and biembeddings. A non-zero sum Heffter array $\mathrm{N}\mathrm{H}(m,n; h,k)$ is an $m \times n$ p. f. array with entries in $\mathbb{Z}_{2nk+1}$ such that: each row contains $h$ filled cells and each column contains $k$ filled cells; for every $x\in \mathbb{Z}_{2nk+1}\setminus\{0\}$, either $x$ or $-x$ appears in the array; the sum of the elements in every row and column is different from $0$ (in $\mathbb{Z}_{2nk+1}$). Here first we explain the connections with relative difference families and with path decompositions of the complete multipartite graph. Then we present a complete solution for the existence problem and a constructive complete solution for the square case and for the rectangular case with no empty cells when the additional, very restrictive, property of "globally simple" is required. Finally, we show how these arrays can be used to construct biembeddings of complete graphs.

math.CO↗

arXiv2022

On the number of non-isomorphic (simple) $k$-gonal biembeddings of complete multipartite graphs

This article aims to provide exponential lower bounds on the number of non-isomorphic $k$-gonal biembeddings of the complete multipartite graph into orientable surfaces. For this purpose, we use the concept, introduced by Archdeacon in 2015, of Heffer array and its relations with graph embeddings. In particular we show that, under certain hypotheses, from a single Heffter array, we can obtain an exponential number of distinct graph embeddings. Exploiting this idea starting from the arrays constructed by Cavenagh, Donovan and Yazici in 2020, we obtain that, for infinitely many values of $k$ and $v$, there are at least $k^{\frac{k}{2}+o(k)} \cdot 2^{v\cdot \frac{H(1/4)}{(2k)^2}+o(v)}$ non-isomorphic $k$-gonal biembeddings of $K_v$, where $H(\cdot)$ is the binary entropy. Moreover about the embeddings of $K_{\frac{v}{t}\times t}$, for $t\in\{1,2,k\}$, we provide a construction of $2^{v\cdot \frac{H(1/4)}{2k(k-1)}+o(v,k)}$ non-isomorphic $k$-gonal biembeddings whenever $k$ is odd and $v$ belongs to a wide infinite family of values.

math.CO↗

arXiv2021

On $λ$-fold relative Heffter arrays and biembedding multigraphs on surfaces

In this paper we define a new class of partially filled arrays, called $λ$-fold relative Heffter arrays, that are a generalisation of the Heffter arrays introduced by Archdeacon in 2015. After showing the connection of this new concept with several other ones, such as signed magic arrays, graph decompositions and relative difference families, we determine some necessary conditions and we present existence results for infinite classes of these arrays. In the last part of the paper we also show that these arrays give rise to biembeddings of multigraphs into orientable surfaces and we provide infinite families of such biembeddings. To conclude, we present a result concerning pairs of $λ$-fold relative Heffter arrays and covering surfaces.

math.CO↗

arXiv2021

Growable Realizations: a Powerful Approach to the Buratti-Horak-Rosa Conjecture

Label the vertices of the complete graph $K_v$ with the integers $\{ 0, 1, \ldots, v-1 \}$ and define the length of the edge between $x$ and $y$ to be $\min( |x-y| , v - |x-y| )$. Let $L$ be a multiset of size $v-1$ with underlying set contained in $\{ 1, \ldots, \lfloor v/2 \rfloor \}$. The Buratti-Horak-Rosa Conjecture is that there is a Hamiltonian path in $K_v$ whose edge lengths are exactly $L$ if and only if for any divisor $d$ of $v$ the number of multiples of $d$ appearing in $L$ is at most $v-d$. We introduce "growable realizations," which enable us to prove many new instances of the conjecture and to reprove known results in a simpler way. As examples of the new method, we give a complete solution when the underlying set is contained in $\{ 1,4,5 \}$ or in $\{ 1,2,3,4 \}$ and a partial result when the underlying set has the form $\{ 1, x, 2x \}$. We believe that for any set $U$ of positive integers there is a finite set of growable realizations that implies the truth of the Buratti-Horak-Rosa Conjecture for all but finitely many multisets with underlying set $U$.

math.CO↗

arXiv2020

New methods to attack the Buratti-Horak-Rosa conjecture

The conjecture, still widely open, posed by Marco Buratti, Peter Horak and Alex Rosa states that a list $L$ of $v-1$ positive integers not exceeding $\left\lfloor \frac{v}{2}\right\rfloor$ is the list of edge-lengths of a suitable Hamiltonian path of the complete graph with vertex-set $\{0,1,\ldots,v-1\}$ if and only if, for every divisor $d$ of $v$, the number of multiples of $d$ appearing in $L$ is at most $v-d$. In this paper we present new methods that are based on linear realizations and can be applied to prove the validity of this conjecture for a vast choice of lists. As example of their flexibility, we consider lists whose underlying set is one of the following: $\{x,y,x+y\}$, $\{1,2,3,4\}$, $\{1,2,4,\ldots,2x\}$, $\{1,2,4,\ldots,2x,2x+1\}$. We also consider lists with many consecutive elements.

math.CO↗

arXiv2020

A reduction of the spectrum problem for odd sun systems and the prime case

A $k$-cycle with a pendant edge attached to each vertex is called a $k$-sun. The existence problem for $k$-sun decompositions of $K_v$, with $k$ odd, has been solved only when $k=3$ or $5$. By adapting a method used by Hoffmann, Lindner and Rodger to reduce the spectrum problem for odd cycle systems of the complete graph, we show that if there is a $k$-sun system of $K_v$ ($k$ odd) whenever $v$ lies in the range $2k< v < 6k$ and satisfies the obvious necessary conditions, then such a system exists for every admissible $v\geq 6k$.

math.CO↗

arXiv2020

Relative Heffter arrays and biembeddings

Relative Heffter arrays, denoted by $\mathrm{H}_t(m,n; s,k)$, have been introduced as a generalization of the classical concept of Heffter array. A $\mathrm{H}_t(m,n; s,k)$ is an $m\times n$ partially filled array with elements in $\mathbb{Z}_v$, where $v=2nk+t$, whose rows contain $s$ filled cells and whose columns contain $k$ filled cells, such that the elements in every row and column sum to zero 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 show how relative Heffter arrays can be used to construct biembeddings of cyclic cycle decompositions of the complete multipartite graph $K_{\frac{2nk+t}{t}\times t}$ into an orientable surface. In particular, we construct such biembeddings providing integer globally simple square relative Heffter arrays for $t=k=3,5,7,9$ and $n\equiv 3 \pmod 4$ and for $k=3$ with $t=n,2n$, any odd $n$.

math.CO↗

arXiv2019

A generalization of Heffter arrays

In this paper we define a new class of partially filled arrays, called relative Heffter arrays, that are a generalization of the Heffter arrays introduced by Archdeacon in 2015. Let $v=2nk+t$ be a positive integer, where $t$ divides $2nk$, and let $J$ be the subgroup of $\mathbb{Z}_v$ of order $t$. A $H_t(m,n; s,k)$ Heffter array over $\mathbb{Z}_v$ relative to $J$ is an $m\times n$ partially filled array with elements in $\mathbb{Z}_v$ such that: (a) each row contains $s$ filled cells and each column contains $k$ filled cells; (b) for every $x\in \mathbb{Z}_v\setminus J$, either $x$ or $-x$ appears in the array; (c) the elements in every row and column sum to $0$. Here we study the existence of square integer (i.e. with entries chosen in $\pm\left\{1,\dots,\left\lfloor \frac{2nk+t}{2}\right\rfloor \right\}$ and where the sums are zero in $\mathbb{Z}$) relative Heffter arrays for $t=k$, denoted by $H_k(n;k)$. In particular, we prove that for $3\leq k\leq n$, with $k\neq 5$, there exists an integer $H_k(n;k)$ if and only if one of the following holds: (a) $k$ is odd and $n\equiv 0,3\pmod 4$; (b) $k\equiv 2\pmod 4$ and $n$ is even; (c) $k\equiv 0\pmod 4$. Also, we show how these arrays give rise to cyclic cycle decompositions of the complete multipartite graph.

math.CO↗

arXiv2019

A tour problem on a toroidal board

In this paper we study a tour problem that we came cross while studying biembeddings and Heffter arrays, see [D.S. Archdeacon, Heffter arrays and biembedding graphs on surfaces, Electron. J. Combin. 22 (2015) #P1.74]. Let $A$ be an $n\times m$ toroidal array consisting of filled cells and empty cells. Assume that an orientation $R=(r_1,\dots,r_n)$ of each row and $C=(c_1,\dots,c_m)$ of each column of $A$ is fixed. 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 $c_{j_{k+1}}$. We propose 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 $A$? Here we provide a complete solution for the case with no empty cells and we obtain partial results for square arrays where the filled cells follow some specific regular patterns.

math.CO↗