arXiv subjects
Publications and source records attributed to Francesca Merola.
We present some results on the harmonious colorings of the Levi graph of a $2$-design, focusing on Steiner $2$-designs. It is easily seen that the harmonious chromatic number of such a Levi graph is at least the number of points of the design: we study and construct Banff designs, that is, designs such that this lower bound is attained.
An $\ell$-cycle decomposition of $K_v$ is said to be \emph{equitably $2$-colourable} if there is a $2$-vertex-colouring of $K_v$ such that each colour is represented (approximately) an equal number of times on each cycle: more precisely, we ask that in each cycle $C$ of the decomposition, each colour appears on $\lfloor \ell/2 \rfloor$ or $\lceil \ell/2 \rceil$ of the vertices of $C$. In this paper we study the existence of equitably 2-colourable $\ell$-cycle decompositions of $K_v$, where $\ell$ is odd, and prove the existence of such a decomposition for $v \equiv 1, \ell$ (mod $2\ell$).
In this paper, we study the existence problem for cyclic $\ell$-cycle decompositions of the graph $K_m[n]$, the complete multipartite graph with $m$ parts of size $n$, and give necessary and sufficient conditions for their existence in the case that $2\ell \mid (m-1)n$.
In this work we introduce Fano Kaleidoscopes, Hesse Kaleidoscopes and their generalizations. These are a particular kind of colored designs for which we will discuss general theory, present some constructions and prove existence results. In particular, using difference methods we show the existence of both a Fano and a Hesse Kaleidoscope on $v$ points when $v$ is a prime or prime power congruent to 1$\pmod{6}$, $v\ne13$. In the Fano case this, together with known results on pairwise balanced designs, allows us to prove the existence of Kaleidoscopes of order $v$ for many other values of $v$; we discuss what the situation is, on the other hand, in the Hesse and general case.
A hamiltonian cycle system (HCS, for short) of a graph $\Gamma$ is a partition of the edges of $\Gamma$ into hamiltonian cycles. A HCS is cyclic when it is invariant under a cyclic permutation of all the vertices of $\Gamma$; the existence problem for a cyclic HCS has been completely solved by Buratti and Del Fra in 2004 when $\Gamma$ is the complete graph $K_v$, $v$ odd, and by Jordon and Morris in 2008 when $\Gamma$ is the complete graph minus a $1$-factor $K_v-I$, $v$ even. In this work we present a complete solution to the existence problem of a cyclic HCS for $\Gamma = K_{m\times n}$, the complete multipartite graph, when the number of parts $m$ is even. We also give necessary and sufficient conditions for the existence of a cyclic and symmetric HCS of $\Gamma$; the notion of a symmetric HCS of a graph $\Gamma$ has been introduced in 2004 by Akiyama, Kobayashi, and Nakamura for $\Gamma =K_v$, $v$ odd, in 2011 by Brualdi and Schroeder when $\Gamma = K_v-I$, $v$ even, and, very recently, by Schroeder when $\Gamma$ is the complete multipartite graph.
It is conjectured that for every pair $(\ell,m)$ of odd integers greater than 2 with $m \equiv 1\; \pmod{\ell}$, there exists a cyclic two-factorization of $K_{\ell m}$ having exactly $(m-1)/2$ factors of type $\ell^m$ and all the others of type $m^{\ell}$. The authors prove the conjecture in the affirmative when $\ell \equiv 1\; \pmod{4}$ and $m \geq \ell^2 -\ell + 1$.