On d-graceful labelings

In this paper we introduce a generalization of the well known concept of a graceful labeling. Given a graph G with e=dm edges, we call d-graceful labeling of G an injective function from V(G) to the set {0,1,2,..., d(m+1)-1} such that {|f(x)-f(y)| | [x,y]\in E(G)} ={1,2,3,...,d(m+1)-1}-{m+1,2(m+1),...,(d-1)(m+1)}. In the case of d=1 and of d=e we find the classical notion of a graceful labeling and of an odd graceful labeling, respectively. Also, we call d-graceful \alpha-labeling of a bipartite graph G a d-graceful labeling of G with the property that its maximum value on one of the two bipartite sets does not reach its minimum value on the other one. We show that these new concepts allow to obtain certain cyclic graph decompositions. We investigate the existence of d-graceful \alpha-labelings for several classes of bipartite graphs, completely solving the problem for paths and stars and giving partial results about cycles of even length and ladders.Comment: In press on Ars Combi

On the existence of elementary abelian cycle systems

We present some necessary and/or sufficient conditions for the existence of an elementary abelian cycle system of the complete graph.We propose a construction for some classes of perfect elementary abelian cycle systems. Finally we consider elementary abelian k-cycle systems of the complete multipartite graph

New results on path-decompositions and their down-links

In a recent paper the concept of \emph{down-link} from a $(K_v,\Gamma)$-design \cB to a $(K_n,\Gamma')$-design \cB' has been introduced. In the present paper the spectrum problems for $\Gamma'=P_4$ are studied. General results on the existence of path-decompositions and embeddings between path-decompositions playing a fundamental role for the construction of down-links are also presented

The seating couple problem in even case

In this paper we consider the seating couple problem with an even number of seats, which, using graph theory terminology, can be stated as follows. Given a positive even integer $v=2n$ and a list $L$ containing $n$ positive integers not exceeding $n$, is it always possible to find a perfect matching of $K_v$ whose list of edge-lengths is $L$? Up to now a (non-constructive) solution is known only when all the edge-lengths are coprime with $v$. In this paper we firstly present some necessary conditions for the existence of a solution. Then, we give a complete constructive solution when the list consists of one or two distinct elements, and when the list consists of consecutive integers $1,2,\ldots,x$, each one appearing with the same multiplicity. Finally, we propose a conjecture and some open problems.Comment: 16 page

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

Effect of manidipine-simvastatin combination on fibrinolysis, adhesion molecules and C-reactive protein in hypertensive hypercholesterolemic patients

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Using Virtual Reality to Rehabilitate Neglect

Purpose: Virtual Reality (VR) platforms gained a lot of attention in the rehabilitation field due to their ability to engage patients and the opportunity they offer to use real world scenarios. As neglect is characterized by an impairment in exploring space that greatly affects daily living, VR could be a powerful tool compared to classical paper and pencil tasks and computer training. Nevertheless, available platforms are costly and obstructive. Here we describe a low cost platform for neglect rehabilitation, that using consumer equipments allows the patient to train at home in an intensive fashion. Method: We tested the platform on IB, a chronic neglect patient, who did not benefit from classical rehabilitation. Results: Our results show that IB improved both in terms of neglect and attention. Importantly, these ameliorations lasted at a follow up evaluation 5 months after the last treatment session and generalized to everyday life activities. Conclusions: VR platforms built using equipment technology and following theoretical principles on brain functioning may induce greater ameliorations in visuo-spatial deficits than classical paradigms possibly thanks to the real world scenarios in association with the "visual feedback" of the patient's own body operating in the virtual environmen

Benthic trophic interactions in an Antarctic shallow water ecosystem affected by recent glacier retreat

The western Antarctic Peninsula is experiencing strong environmental changes as a consequence of ongoing regional warming. Glaciers in the area are retreating rapidly and increased sediment-laden meltwater runoff threatens the benthic biodiversity at shallow depths. We identified three sites with a distinct glacier-retreat related history and different levels of glacial influence in the inner part of Potter Cove (King George Island, South Shetland Islands), a fjord-like embayment impacted since the 1950s by a tidewater glacier retreat. We compared the soft sediment meio- and macrofauna isotopic niche widths (d13C and d15N stable isotope analysis) at the three sites to investigate possible glacier retreat-related influences on benthic trophic interactions. The isotopic niches were locally shaped by the different degrees of glacier retreat-related disturbance within the Cove. Wider isotopic niche widths were found at the site that has become ice-free most recently, and narrower niches at the older ice-free sites. At an intermediate state of glacier retreat-related disturbance (e.g. via ice-growler scouring) species with different strategies could settle. The site at the earliest stage of post-retreat development was characterized by an assemblage with lower trophic redundancy. Generally, the isotopic niche widths increased with increasing size spectra of organisms within the community, excepting the youngest assemblage, where the pioneer colonizer meiofauna size class displayed the highest isotopic niche width. Meiofauna at all sites generally occupied positions in the isotopic space that suggested a detrital-pool food source and/or the presence of predatory taxa. In general ice scour and glacial impact appeared to play a two-fold role within the Cove: i) either stimulating trophic diversity by allowing continuous re-colonization of meiofaunal species or, ii) over time driving the benthic assemblages into a more compact trophic structure with increased connectedness and resource recycling