Classifying Coloring Graphs

Given a graph G, its k-coloring graph is the graph whose vertex set is the proper k-colorings of the vertices of G with two k-colorings adjacent if they differ at exactly one vertex. In this paper, we consider the question: Which graphs can be coloring graphs? In other words, given a graph H, do there exist G and k such that H is the k-coloring graph of G? We will answer this question for several classes of graphs and discuss important obstructions to being a coloring graph involving order, girth, and induced subgraphs

La Salle Faculty Panel: Questions, Problems and Approaches to Working with La Salle Students

The La Salle Summer Workshop Program (June, 2012) included a Faculty Panel with representatives from several of La Salle\u27s professional programs and the mathematics and science departments. Here are some of their comments

N-terminal B-type natriuretic peptide urinary concentrations and retinopathy of prematurity

BACKGROUND: To validate the findings of a single-center pilot study showing elevated urinary N-terminal B-type natriuretic peptide (NTproBNP) concentrations in preterm infants subsequently developing severe retinopathy of prematurity (ROP) in a multicenter setting across eight European and Middle East countries. METHODS: Prospective observational study in 967 preterm infants <30 weeks' gestational age assessing the capacity of urinary NTproBNP on days of life (DOLs) 14 and 28 to predict ROP requiring treatment. RESULTS: Urinary NTproBNP concentrations were markedly elevated in infants who developed ROP requiring treatment (n = 94) compared with survivors without ROP treatment (n = 837), at both time points (median (interquartile range) DOL14: 8,950 (1,925-23,783) vs. 3,083 (1,193-17,393) vs. 816 (290-3,078) pg/ml, P <0.001) and DOL28 (2,203 (611-4,063) vs. 1,671 (254-11,340) vs. 408 (162-1,126) pg/ml, P <0.001). C-statistic of NTproBNP for treated ROP or death was 0.731 (95% confidence interval 0.654-0.774) for DOL14 and 0.683 (0.622-0.745) for DOL28 (P <0.001). Threshold scores were calculated, potentially enabling around 20% of infants with low NTproBNP scores never to be screened with ophthalmoscopy. CONCLUSION: There is a strong association between early urinary NTproBNP and subsequent ROP development, which can be used to further refine subgroups of patients with high or low risk of severe RO