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

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

Comparison of clinical and angiographic prognostic risk scores in elderly patients presenting with acute coronary syndrome and referred for percutaneous coronary intervention.

BACKGROUND: Multiple risk prediction models have been validated in all-age patients presenting with acute coronary syndrome (ACS) and treated with percutaneous coronary intervention (PCI); however, they have not been validated specifically in the elderly. METHODS: We calculated the GRACE (Global Registry of Acute Coronary Events) score, the logistic EuroSCORE, the AMIS (Acute Myocardial Infarction Swiss registry) score, and the SYNTAX (Synergy between Percutaneous Coronary Intervention with TAXUS and Cardiac Surgery) score in a consecutive series of 114 patients ≥75 years presenting with ACS and treated with PCI within 24 hours of hospital admission. Patients were stratified according to score tertiles and analysed retrospectively by comparing the lower/mid tertiles as an aggregate group with the higher tertile group. The primary endpoint was 30-day mortality. Secondary endpoints were the composite of death and major adverse cardiovascular events (MACE) at 30 days, and 1-year MACE-free survival. Model discrimination ability was assessed using the area under receiver operating characteristic curve (AUC). RESULTS: Thirty-day mortality was higher in the upper tertile compared with the aggregate lower/mid tertiles according to the logistic EuroSCORE (42% vs 5%; odds ratio [OR] = 14, 95% confidence interval [CI] = 4-48; p <0.001; AUC = 0.79), the GRACE score (40% vs 4%; OR = 17, 95% CI = 4-64; p <0.001; AUC = 0.80), the AMIS score (40% vs 4%; OR = 16, 95% CI = 4-63; p <0.001; AUC = 0.80), and the SYNTAX score (37% vs 5%; OR = 11, 95% CI = 3-37; p <0.001; AUC = 0.77). CONCLUSIONS: In elderly patients presenting with ACS and referred to PCI within 24 hours of admission, the GRACE score, the EuroSCORE, the AMIS score, and the SYNTAX score predicted 30 day mortality. The predictive value of clinical scores was improved by using them in combination

GRB 081028 and its late-time afterglow re-brightening

‘The definitive version is available at www3.interscience.wiley.com '. Copyright Royal Astronomical SocietySwift captured for the first time a smoothly rising X-ray re-brightening of clear non-flaring origin after the steep decay in a long gamma-ray burst (GRB): GRB 081028. A rising phase is likely present in all GRBs but is usually hidden by the prompt tail emission and constitutes the first manifestation of what is later to give rise to the shallow decay phase. Contemporaneous optical observations reveal a rapid evolution of the injection frequency of a fast cooling synchrotron spectrum through the optical band, which disfavours the afterglow onset (start of the forward shock emission along our line of sight when the outflow is decelerated) as the origin of the observed re-brightening. We investigate alternative scenarios and find that the observations are consistent with the predictions for a narrow jet viewed off-axis. The high on-axis energy budget implied by this interpretation suggests different physical origins of the prompt and (late) afterglow emission. Strong spectral softening takes place from the prompt to the steep decay phase: we track the evolution of the spectral peak energy from the γ-rays to the X-rays and highlight the problems of the high latitude and adiabatic cooling interpretations. Notably, a softening of both the high and low spectral slopes with time is also observed. We discuss the low on-axis radiative efficiency of GRB 081028 comparing its properties against a sample of Swift long GRBs with secure Eγ,iso measurements.Peer reviewe