arXiv subjects

T. Traetta

Publications and source records attributed to T. Traetta.

7 recordsLinked to original sources

arXiv2025

On circular external difference families

Circular external difference families (CEDFs) are a recently-introduced variation of external difference families (EDFs) with applications to non-malleable threshold schemes: a $(v,m,\ell,1)$-CEDF is an $m$-sequence $(A_0, \ldots, A_{m-1})$ of $\ell$-subsets of an additive group $G$ of order $v$ such that $G\setminus\{0\}$ equals the multiset of all differences $a-a'$, with $(a,a')\in A_{i+1}\times A_{i}$ for some $i \in \mathbb{Z}_m$. When $G$ is the cyclic group, we speak of a cyclic CEDF. The existence of cyclic $(v,m,\ell,1)$-CEDFs is well understood when $m$ is even, while nonexistence is known when both $m$ and $\ell$ are odd. However, the case where $m$ is odd and $\ell$ is even has only been resolved in a few special cases. In this paper, we address this gap by constructing cyclic $(v,m,\ell,1)$-CEDFs for any odd $m>1$ when $\ell=2$, and for any even $\ell \ge 2$ when $m=3$. Notably, the latter result relies on the existence of a suitable tiling of the multiplicative semigroup of $\mathbb{Z}_v\setminus\{0\}$. Moreover, noting that every $(v,3,\ell,1)$-CEDF produces a $(v,3,\ell,2)$-EDF, we completely solve the existence problem for $(3\ell^2+1,3,\ell,2)$-EDFs over an abelian group. Our approach is based on representing the blocks as arithmetic progressions and analyzing their step patterns. We present two different ways to construct cyclic $(v,m,2,1)$-CEDFs for every odd $m>1$; their step patterns show that the resulting CEDFs are inequivalent. Many additional inequivalent CEDFs are obtained by translating suitable subsets within the CEDF.

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arXiv2024

A sharp upper bound for the harmonious total chromatic number of graphs and multigraphs

A proper total colouring of a graph $G$ is called harmonious if it has the further property that when replacing each unordered pair of incident vertices and edges with their colours, then no pair of colours appears twice. The smallest number of colours for it to exist is called the harmonious total chromatic number of $G$, denoted by $h_t(G)$. Here, we give a general upper bound for $h_t(G)$ in terms of the order $n$ of $G$. Our two main results are obvious consequences of the computation of the harmonious total chromatic number of the complete graph $K_n$ and of the complete multigraph $\lambda K_n$, where $\lambda$ is the number of edges joining each pair of vertices of $K_n$. In particular, Araujo-Pardo et al. have recently shown that $\frac{3}{2}n\leq h_t(K_n) \leq \frac{5}{3}n +\theta(1)$. In this paper, we prove that $h_t(K_{n})=\left\lceil \frac{3}{2}n \right\rceil$ except for $h_t(K_{1})=1$ and $h_t(K_{4})=7$; therefore, $h_t(G) \le \left\lceil \frac{3}{2}n \right\rceil$, for every graph $G$ on $n>4$ vertices. Finally, we extend such a result to the harmonious total chromatic number of the complete multigraph $\lambda K_n$ and as a consequence show that $h_t(\mathcal{G})\leq (\lambda-1)(2\left\lceil\frac{n}{2}\right\rceil-1)+\left\lceil\frac{3n}{2}\right\rceil$ for $n>4$, where $\mathcal{G}$ is a multigraph such that $\lambda$ is the maximum number of edges between any two vertices.

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arXiv2022

On non-isomorphic biminimal pots realizing the cube

In this paper, we disprove a conjecture recently proposed in [L. Almodovar et al., arXiv:2108.00035] on the non-existence of biminimal pots realizing the cube, namely pots with the minimum number of tiles and the minimum number of bond-edge types. In particular, we present two biminimal pots realizing the cube and show that these two pots are unique up to isomorphisms.

math.CO↗

arXiv2022

Constructing uniform 2-factorizations via row-sum matrices: solutions to the Hamilton-Waterloo problem

In this paper, we formally introduce the concept of a row-sum matrix over an arbitrary group $G$. When $G$ is cyclic, these types of matrices have been widely used to build uniform 2-factorizations of small Cayley graphs (or, Cayley subgraphs of blown-up cycles), which themselves factorize complete (equipartite) graphs. Here, we construct row-sum matrices over a class of non-abelian groups, the generalized dihedral groups, and we use them to construct uniform $2$-factorizations that solve infinitely many open cases of the Hamilton-Waterloo problem, thus filling up large parts of the gaps in the spectrum of orders for which such factorizations are known to exist.

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arXiv2020

On the Oberwolfach problem for single-flip $2$-factors via graceful labelings

Let $F$ be a $2$-regular graph of order $v$. The Oberwolfach problem $OP(F)$, posed in 1967 and still open, asks for a decomposition of $K_v$ into copies of $F$. In this paper we show that $OP(F)$ has a solution whenever $F$ has a sufficiently large cycle which meets a given lower bound and, in addition, has a single-flip automorphism, which is an involutory automorphism acting as a reflection on exactly one of the cycles of $F$. Furthermore, we prove analogous results for the minimum covering version and the maximum packing version of the problem. We also show a similar result when the edges of $K_v$ have multiplicity 2, but in this case we do not require that $F$ be single-flip. Our approach allows us to explicitly construct solutions to the Oberwolfach Problem with well-behaved automorphisms, in contrast with some recent asymptotic results, based on probabilistic methods, which are nonconstructive and do not provide a lower bound on the order of $F$ that guarantees the solvability of $OP(F)$. Our constructions are based on a doubling construction which applies to graceful labelings of $2$-regular graphs with a vertex removed. We show that this class of graphs is graceful as long as the length of the path-component is sufficiently large. A much better lower bound on the length of the path is given for an $\alpha$-labeling of such graphs to exist.

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arXiv2018

The Hamilton-Waterloo Problem with even cycle lengths

The Hamilton-Waterloo Problem HWP$(v;m,n;\alpha,\beta)$ asks for a 2-factorization of the complete graph $K_v$ or $K_v-I$, the complete graph with the edges of a 1-factor removed, into $\alpha$ $C_m$-factors and $\beta$ $C_n$-factors, where $3 \leq m < n$. In the case that $m$ and $n$ are both even, the problem has been solved except possibly when $1 \in \{\alpha,\beta\}$ or when $\alpha$ and $\beta$ are both odd, in which case necessarily $v \equiv 2 \pmod{4}$. In this paper, we develop a new construction that creates factorizations with larger cycles from existing factorizations under certain conditions. This construction enables us to show that there is a solution to HWP$(v;2m,2n;\alpha,\beta)$ for odd $\alpha$ and $\beta$ whenever the obvious necessary conditions hold, except possibly if $\beta=1$; $\beta=3$ and $\gcd(m,n)=1$; $\alpha=1$; or $v=2mn/\gcd(m,n)$. This result almost completely settles the existence problem for even cycles, other than the possible exceptions noted above.

math.CO↗

arXiv2015

On the Hamilton-Waterloo Problem with odd orders

Given non-negative integers $v, m, n, \alpha, \beta$, the Hamilton-Waterloo problem asks for a factorization of the complete graph $K_v$ into $\alpha$ $C_m$-factors and $\beta$ $C_n$-factors. Clearly, $v$ odd, $n,m\geq 3$, $m\mid v$, $n\mid v$ and $\alpha+\beta = (v-1)/2$ are necessary conditions. To date results have only been found for specific values of $m$ and $n$. In this paper we show that for any $m$ and $n$ the necessary conditions are sufficient when $v$ is a multiple of $mn$ and $v>mn$, except possibly when $\beta=1$ or 3, with five additional possible exceptions in $(m,n,\beta)$. For the case where $v=mn$ we show sufficiency when $\beta > (n+5)/2$ except possibly when $(m,\alpha) = (3,2)$, $(3,4)$, with seven further possible exceptions in $(m,n,\alpha,\beta)$. We also show that when $n\geq m\geq 3$ are odd integers, the lexicographic product of $C_m$ with the empty graph of order $n$ has a factorization into $\alpha$ $C_m$-factors and $\beta$ $C_n$-factors for every $0\leq \alpha \leq n$, $\beta = n-\alpha$, except possibly when $\alpha= 2,4$, $\beta = 1, 3$, with three additional possible exceptions in $(m,n,\alpha)$.

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