Star 5-edge-colorings of subcubic multigraphs

The star chromatic index of a multigraph $G$, denoted $\chi'_{s}(G)$, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length four is bi-colored. A multigraph $G$ is star $k$-edge-colorable if $\chi'_{s}(G)\le k$. Dvo\v{r}\'ak, Mohar and \v{S}\'amal [Star chromatic index, J Graph Theory 72 (2013), 313--326] proved that every subcubic multigraph is star $7$-edge-colorable, and conjectured that every subcubic multigraph should be star $6$-edge-colorable. Kerdjoudj, Kostochka and Raspaud considered the list version of this problem for simple graphs and proved that every subcubic graph with maximum average degree less than $7/3$ is star list-$5$-edge-colorable. It is known that a graph with maximum average degree $14/5$ is not necessarily star $5$-edge-colorable. In this paper, we prove that every subcubic multigraph with maximum average degree less than $12/5$ is star $5$-edge-colorable.Comment: to appear in Discrete Mathematics. arXiv admin note: text overlap with arXiv:1701.0410

Roles of TGFβ and FGF signals during growth and differentiation of mouse lens epithelial cell in vitro.

Transforming growth factor β (TGFβ) and fibroblast growth factor (FGF) signaling pathways play important roles in the proliferation and differentiation of lens epithelial cells (LECs) during development. Low dosage bFGF promotes cell proliferation while high dosage induces differentiation. TGFβ signaling regulates LEC proliferation and differentiation as well, but also promotes epithelial-mesenchymal transitions that lead to cataracts. Thus far, it has been difficult to recapitulate the features of germinative LECs in vitro. Here, we have established a LEC culture protocol that uses SB431542 (SB) compound to inhibit TGFβ/Smad activation, and found that SB treatment promoted mouse LEC proliferation, maintained LECs' morphology and distinct markers including N-cadherin, c-Maf, Prox1, and αA-, αB-, and β-crystallins. In contrast, low-dosage bFGF was unable to sustain those markers and, combined with SB, altered LECs' morphology and β-crystallin expression. We further found that Matrigel substrate coatings greatly increased cell proliferation and uniquely affected β-crystallin expression. Cultured LECs retained the ability to differentiate into γ-crystallin-positive lentoids by high-dosage bFGF treatment. Thus, a suppression of TGFβ/Smad signaling in vitro is critical to maintaining characteristic features of mouse LECs, especially expression of the key transcription factors c-Maf and Prox1