Rainbow Pancyclicity in a Collection of Graphs Under the Dirac-type Condition

Let G={Gi:i∈[n]} be a collection of not necessarily distinct n-vertex graphs with the same vertex set V,where G can be seen as an edge-colored(multi)graph and each Gi is the set of edges with color i.A graph F on V is called rainbow if any two edges of F come from different Gis’.We say that G is rainbow pancyclic if there is a rainbow cycle Cℓof lengthℓin G for each integerℓ2[3,n].In 2020,Joos and Kim proved a rainbow version of Dirac’s theorem:Ifδ(Gi)≥2/n for each i∈[n],then there is a rainbow Hamiltonian cycle in G.In this paper,under the same condition,we show that G is rainbow pancyclic except that n is even and G consists of n copies of Kn/2,n/2.This result supports the famous meta-conjecture posed by Bondy.

Role of nesfatin-1 in a rat model of visceral hypersensitivity

AIM: To explore the role of nesfatin-1 on irritable bowel syndrome (IBS)-like visceral hypersensitivity. METHODS: The animal model of IBS-like visceral hypersensitivity was induced by intracolonic infusion of 0.5% acetic acid (AA) in saline once daily from postnatal days 8-21. Experiments were performed when rats became adults. The visceral sensitivity of rats was evaluated by abdominal withdrawal reflex (AWR) and electromyographic (EMG) activity of the external oblique muscle to graded colorectal distension. The content of nesfatin-1 in serum was determined using enzyme-linked immunosorbent assay. After implantation of an intracerebroventricular (ICV) cannula and two electrodes into the external oblique muscle, model rats were randomly divided into four groups. Animals then received ICV injection of 8 μg of anti-nesfatin-1/ nucleobindin-2 (NUCB2), 50 μg of α-helical cortico-tropin releasing factor (CRF) 9-41 (non-selective CRF receptor antagonist), 50 μg of NBI-27914 (selective CRF1 receptor antagonist) or 5 μL of vehicle. After 1 h of ICV administration, visceral sensitivity of each group was measured again, and comparisons between groups were made. RESULTS: Rats treated with AA showed higher mean AWR scores and EMG activity at all distension pressures compared with controls (P < 0.05). On histopathologic examination, no evidence of inflammation or abnormalities in structure were noted in the colon of either control or AA-treated groups. Myeloperoxidase values were not significantly different between the two groups. The level of nesfatin-1 in serum was significantly higher in the AA-treated group than in the control group (5.34 ± 0.37 ng/mL vs 4.81 ± 0.42 ng/mL, P < 0.01). Compared with rats injected with vehicle, rats which received ICV anti-nesfatin-1/NUCB2, α-helical CRF9-41 or NBI-27914 showed decreased mean AWR scores and EMG activity at all distension pressures (P < 0.05). CONCLUSION: Nesfatin-1 may be associated with IBS-like visceral hypersensitivity, which may be implicated in brain CRF/CRF1 signaling pathways.

Epidermal growth factor upregulates serotonin transporter and its association with visceral hypersensitivity in irritable bowel syndrome

AIM: To investigate the role of epidermal growth factor (EGF) in visceral hypersensitivity and its effect on the serotonin transporter (SERT).

Rainbow k-connectivity of Random Bipartite Graphs

A path in an edge-colored graph G is called a rainbow path if no two edges of the path are colored the same color.The minimum number of colors required to color the edges of G such that every pair of vertices are connec ted by at least k internally ver tex-disjoint rainbow paths is called the rainbow k-connectivity of the graph G,denoted by rck(G).For the random graph G(n,p),He and Liang got a sharp threshold function for the property rck(G(n,p))≤d.For the random equi-bipartite graph G(n,n,p),Fujita et.al.got a sharp threshold function for the property rck(G(n,n,p))≤3.They also posed the following problem:For d≥2,determine a sharp threshold function for the property rck(G)≤d,where G is another random graph model.This paper is to give a solution to their problem in the general random bipartite graph model G(m,n,p).

Upper Bound Involving Parameter σ_2 for the Rainbow Connection Number

Let G be a connected graph of order n. The rainbow connection number rc（G） of G was introduced by Chartrand et al. Chandran et al. used the minimum degree of G and obtained an upper bound that rc（G） 〈_ 3n/（ δ＋ 1） - 3, which is tight up to additive factors. In this paper, we use the minimum degree-sum a2 6n of G to obtain a better bound rc（G） _〈 - 8, especially when is small （constant） but a2 is large （linear in n）.

Partitioning Complete Graphs by Heterochromatic Trees

A heterochromatie tree is an edge-colored tree in which any two edges have different colors. The heterochromatic tree partition number of an r-edge-colored graph G, denoted by tr （G）, is the minimum positive integer p such that whenever the edges of the graph G are colored with r colors, the vertices of G can be covered by at most p vertex-disjoint heterochromatic trees. In this paper we determine the heterochromatic tree partition number of r-edge-colored complete graphs. We also find at most tr（Kn） vertex-disjoint heterochromatic trees to cover all the vertices in polynomial time for a given r-edge-coloring of Kn.

Hypergraph Turan Numbers of Vertex Disjoint Cycles

The Turan number of a k-uniform hypergraph H,denoted by exk(n;H),is the maximum number of edges in any k-uniform hypergraph F on n vertices which does not contain H as a subgraph.Let Cl(k)denote the family of all k-uniform minimal cycles of length l;S(l1,…,lr)denote the family of hypergraphs consisting of unions of r vertex disjoint minimal cycles of lengthl1,…lr,respectively,and Cl(k)denote a k-uniform linear cycle of length l.We determine precisely exk(n;S(l1,…,lr)and exk(n;Cl1(k),…,Cl1(k)for sufficiently large n.Our results extend recent results of Füredi and Jiang who determined the Turan numbers for single k-uniform minimal cycles and linear cycles.

Molecular mechanism of quercitrin on osteogenic differentiation and adipogenic differentiation of rat bone marrow stromal stem cells(rBMSCs)

Objective：The study was designed to investigate the molecular mechanism of quercitrin on osteogenic differentiation and adipogenic differentiation of r BMSCs.Methods：r BMSCs were harvested from SD rats,and determination of alkaline phosphatase（ALP）activity,quantification of mineralization by Alizarin Red S staining,and the m RNA expression of osteogenic differentiation markers（Runx2,BMP-2,and OSX）by RT-PCR after r BMSCs stimulated by osteogenic induction with（0.1–10）μg/m L of quercitrin,quantification of Lipid droplet by Oil Red O staining and the m RNA expression of adipogenic differentiation marker（,and a P2）by RT-PCR after r BMSCs stimulated by adipogenic induction with（0.1-10）μg/m L of quercitrin.Results：Quercitrin can up-regulate the m RNA expression of osteogenic differentiation markers（Runx2,BMP-2,and OSX）and increase ALP activity and mineralization after osteogenic induction,on the other hand quercitrin can suppress the m RNA expression of adipogenic differentiation markers（,and a P2）and decrease lipid droplet after adipogenic induction.Conclusion：This study suggested that quercitrin not only stimulated osteogenic differentiation but also inhibited adipogenic differentiation of r BMSCs,which was associated with the up-regulation of Runx2,BMP-2,and OSX m RNA expression and the down-regulation of,and a P2 m RNA expression.