Search

A graph G is said to be 2-distinguishable if there is a labeling of the vertices with two labels so that only the trivial automorphism preserves the labels. The minimum size of a label class, over all 2-distinguishing labelings, is called the cost of 2-distinguishing, denoted by ρ ( G ) . For n ≥ 4 the hypercubes Q n are 2-distinguishable, but the values for ρ ( Q n ) have been elusive, with only bounds and partial results previously known. This paper settles the question. The main result can be summarized as: for n ≥ 4 , ρ ( Q n ) ∈ { 1 + ⌈ log 2 ⁡ n ⌉ , 2 + ⌈ log 2 ⁡ n ⌉ } . Exact values are found using a recursive relationship involving a new parameter ν m , the smallest integer for which ρ ( Q ν m ) = m . The main result is 4 ≤ n ≤ 12 ⟹ ρ ( Q n ) = 5 , and 5 ≤ m ≤ 11 ⟹ ν m = 4 ; for m ≥ 6 , ρ ( Q n ) = m ⇔ 2 m − 2 − ν m − 1 + 1 ≤ n ≤ 2 m − 1 − ν m ; for n ≥ 5 , ν m = n ⇔ 2 n − 1 − ρ ( Q n − 1 ) + 1 ≤ m ≤ 2 n − ρ ( Q n ) .

Introduction We have recently reported the crucial role of Galectin-3 (Gal3) in the progression of type 3 cardio-renal syndrome (CRS-3). Indeed, Gal3 overexpression after renal ischemia reperfusion (IRr) was associated with cardiac fibrosis and dysfunction, which were prevented by Gal3 pharmacological or genetic inhibition. Objective Our present aim was to better define the initial role of kidney Gal3 for triggering CRS-3. Method A murine irradiated and graft model having IRr was used to differentiate the Gal3 expressed by tissue (WT irradiated mice grafted with Gal3-/- bone marrow (BM) cells, WTGal3-/-BM) from Gal3 expressed by immune cells (Gal3-/- irradiated mice grafted with WT BM cells: Gal3-/-WTBM) and were killed after 2 or 28 days. Cardiac and renal phenotypes were evaluated using echocardiography and real-time PCR. Results After IRr, WTGal3-/-BM exhibit normal cardiac phenotype whereas Gal3-/-WTBM developed cardiac dysfunction and high Fibronectin (Fn) mRNA expression ( Fig. 1 A). In contrast, WTGal3-/-BM showed short- and long-term kidney damages with increased renal NGAL mRNA level by day 2 and increased Fn mRNA by 28d ( Fig. 1 B). In kidney after IRr, the increased levels of endothelial marker transcripts (ICAM-1, VCAM-1) mainly in WTGal3-/-BM mice suggested that tissue Gal3 expression is involved in the endothelial activation. Conclusion After IRr the organ dysfunction varies according to the Gal3 cell type origin; kidney Gal3 plays a major role in renal damage, whereas Gal3 from infiltrated BM cells in the heart is deleterious for cardiac function. It remains now to define how tissue Gal3 acts and interacts with endothelial cells, to assess Gal3 as a key-point for CRS-3 understanding.

A graph has thickness t if the edges can be decomposed into t and no fewer planar layers. We study one aspect of a generalization of Ringel's famous Earth-Moon problem: what is the largest chromatic number of any thickness-2 graph? In particular, given a graph G we consider the r-inflation of G and find bounds on both the thickness and the chromatic number of the inflated graphs. In some instances, the best possible bounds on both the chromatic number and thickness are achieved. We end with several open problems.

A set of points W in Euclidean space is said to realize the finite group G if the isometry group of W is isomorphic to G. We show that every finite group G can be realized by a finite subset of some Rn, with n

Let α∗ denote the maximum number of independent vertices all of which have the same degree. We provide lower bounds for α∗ for graphs that are planar, maximal planar, of bounded degree, or trees.

The epithelium of the mammalian vagina arises from two distinct germ layers, endoderm from the urogenital sinus and mesoderm from the Mullerian ducts. While neonatal vaginal epithelium can be induced to form prostate which is normally an endodermal derivative, it has not been determined whether this ability to form prostate is shared by both mesoderm- and endo-derm-derived vaginal epithelia. To test the competence of vaginal epithelia we have isolated sinus-derived and Miillerian-derived vaginal epithelia from newborn mice, combined them with rat urogenital sinus mesenchyme, and grown the tissue recombinants for 4 weeks in male athymic nude mice. Endoderm-derived sinus vaginal epithelium was induced to form prostatic tissue which expressed prostate-specific secretory proteins in 21 of 23 tissue recombinants. Mullerian-derived vaginal epithelium formed small ducts and cysts lined by a simple epithelium. These latter tissue recombinants lacked any evidence of prostatic secretory proteins. Similarly, endo-derm-derived urethral epithelium was induced to form prostate (17 of 17 cases), while mesoderm-derived uterine epithelium was not (0 of 13 cases). Therefore, the ability to form prostatic epithelium was limited to endodermal derivatives of the urogenital tract.

The cell surface proteoglycan, syndecan, exhibits molecular and histological dimorphism in the mouse uterus and vagina. In the mature vagina, syndecan is localized at the surfaces of the basal and intermediate cells of the stratified epithelium and has a modal molecular mass of ca. 92 kDa. The uterus expresses a larger form of syndecan (ca. 110 kDa) which is detected at the basolateral surfaces of the simple columnar epithelial cells. We have investigated whether epithelial-mesenchymal interactions influence the expression of syndecan in these organs by analyzing tissue recombinants composed of mouse epithelium and rat mesenchyme or vice versa with monoclonal antibody 281-2, which recognizes mouse syndecan. In tissue recombinants composed of newborn mouse uterine epithelium and rat vaginal stroma, the uterine epithelium was induced to form a stratified vaginal epithelium which expressed syndecan in same the pattern and mass typical of vaginal epithelium. Likewise, rat uterine stroma induced newborn mouse vaginal epithelium to undergo uterine development, and this epithelium exhibited a uterine pattern of syndecan expression. Although stromal cells normally express little syndecan in most adult organs, analysis of recombinants composed of mouse stroma and rat epithelium revealed that both uterine and vaginal mouse stromata synthesized syndecan that was larger (ca. 170–190 kDa) than the epithelial syndecans. A quantitative increase in the amount of stromal syndecan was evident when stroma was grown in association with epithelium in comparison to stroma grown by itself. These data suggest that epithelial-mesenchymal interactions influence the amount, localization, and mass of both epithelial and stromal syndecan.

Stages 20 and 25 chick apical ectodermal ridge have been cultured in nutrient medium containing fetal bovine serum and the tissues have been examined for dying cells at 0, 6, 12, 18, and 24 hr. By 12 hr, an average of 43% of the cells were dying. By 24 hr, stage 20 ridge had lost its integrity and stage 25 ridge contained an average of 50% dying cells. These results are in agreement with the observations of R. L. Searls and E. Zwilling (1964, Dev. Biol. 9, 38-55) on isolated stage 20 ridge. In subsequent experiments, ridge ectoderm was cultured in serum-containing medium to which insulin (5 micrograms/ml), transferrin (5 micrograms/ml), and selenium (5 ng/ml) or insulin (5 micrograms/ml) had been added. Under these conditions the ectoderms remained viable even after 24 hr in vitro.