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For positive integers k≥2 and m, let gk(m) be the smallest integer such that for each graph G with m edges there exists a k-partition V(G)=V1∪…∪Vk in which each Vi contains at most gk(m) edges. Bollobas and Scott showed that gk(m)≤mk2+k−12k22m+1/4−1/2. Ma and Yu posed the following problem: is it true that the limsup of mk2+k−12k22m−gk(m) tends to infinity as m tends to infinity? They showed it holds when k is even, establishing a conjecture of Bollobas and Scott. In this article, we solve the problem completely. We also present a result by showing that every graph with a large k-cut has a k-partition in which each vertex class contains relatively few edges, which partly improves a result given by Bollobas and Scott.

In 2002, Bollobas and Scott posed the following problem: for an integer k≥2 and a graph G of m edges, what is the smallest f(k, m) such that V(G) can be partitioned into V 1,…,Vk in which e(Vi∪Vj)≤f(k,m) for all 1≤i≠j≤k, where e(Vi∪Vj) denotes the number of edges with both ends in Vi∪Vj? In this paper, we solve this problem asymptotically by showing that f(k,m)≤m/(k−1)+o(m). We also show that V(G) can be partitioned into V1,…,Vk such that e(Vi∪Vj)≤4m/k2+4Δ/k+o(m) for 1≤i≠j≤k, where Δ denotes the maximum degree of G. This confirms a conjecture of Bollobas and Scott. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 59–70, 2017

Background and Objectives: Bone marrow derived mesenchymal stem cells (BMSCs) have shown to be an appealing source for cell therapy and tissue engineering. Previous studies have confirmed that the application of low-level laser irradiation (LLLI) could affect the cellular process. However, little is known about the effects of LLLI on BMSCs. The aim of this study was designed to investigate the influence of LLLI at different energy densities on BMSCs proliferation, secretion and myogenic differentiation. Study Design/Materials and Methods: BMSCs were harvested from rat fresh bone marrow and exposed to a 635 nm diode laser (60 mW; 0, 0.5, 1.0, 2.0, or 5.0 J/cm 2 ). The lactate dehydrogenase (LDH) release was used to assess the cytotoxicity of LLLI at different energy densities. Cell proliferation was evaluated by using 3-(4, 5-dimethylithiazol-2-yl)-2, 5-diphenyl tetrazolium bromide (MTT) and 5-bromo-2 0 -deoxyuridine (BrdU) assay. Production of vascular endothelial growth factor (VEGF) and nerve growth factor (NGF) were measured by enzyme-linked immunosorbent assay (ELISA). Myogenic differentiation, induced by 5-azacytidine (5-aza), was assessed by using immunocytochemical staining for the expression of sarcomeric a-actin and desmin. Results: Cytotoxicity assay showed no significant difference between the non-irradiated group and irradiated groups. LLLI significantly stimulated BMSCs proliferation and 0.5 J/cm 2 was found to be an optimal energy density. VEGF and NGF were identified and LLLI at 5.0 J/cm 2 significantly stimulated the secretion. After 5-aza induction, myogenic differentiation was observed in all groups and LLLI at 5.0 J/cm 2 dramatically facilitated the differentiation. Conclusions: LLLI stimulates proliferation, increases growth factors secretion and facilitates myogenic differentiation of BMSCs. Therefore, LLLI may provide a novel approach for the preconditioning of BMSCs in vitro prior to transplantation. Lasers Surg. Med. 40:726–733, 2008. 2008 Wiley-Liss, Inc.

One of the most important indicators of myocardial regeneration is cardiomyocyte proliferation. However, it is difficult to distinguish cardiomyocytes in the regenerating stage from binucleated or multinucleated myocytes by conventional morphometric techniques. As cell cycle progression (CCP) scores have been successfully applied to the evaluation of the proliferation of cancer cells, we sought to establish a multi-gene score to evaluate cardiomyocyte proliferation in this study. Given the disturbances of nuclear division without cell division that occurs in cardiomyocytes, ten cytokinesis-correlated genes (Anln, Aurkb, Cenpa, Kif4, Kif23, Prc1, RhoA, Spin1, TACC2, and CDC42) were chosen to establish the cardiomyocyte cytokinesis score (CC-Score). The expression levels of these genes in H9C2 rat cardiomyoblast cells, the proliferation of which were stimulated or inhibited, were detected using qRT-PCR. To confirm the feasibility of the CC-Score system, four conventional methods for evaluating cardiomyocyte proliferation, including the MTT assay, BrdU assay, immunofluorescence, and flow cytometry analysis, were used in each group. The results of the CC-Score in the assessment of the proliferation of H9C2 cells were consistent with those of four commonly used proliferative assay methods. We conclude that the CC-Score can be used to assess the proliferation status of H9C2 cells, and suggest that the CC-Score may be a potential method for the assessment of cardiomyocyte proliferation in myocardial regeneration. However, validation studies utilizing primary cultured rat cardiomyocytes and heart tissue are warranted.