Proof of the Goldberg-Seymour Conjecture on Edge-Colorings of Multigraphs

Given a multigraph $G=(V,E)$, the {\em edge-coloring problem} (ECP) is to color the edges of $G$ with the minimum number of colors so that no two adjacent edges have the same color. This problem can be naturally formulated as an integer program, and its linear programming relaxation is called the {\em fractional edge-coloring problem} (FECP). In the literature, the optimal value of ECP (resp. FECP) is called the {\em chromatic index} (resp. {\em fractional chromatic index}) of $G$, denoted by $\chi'(G)$ (resp. $\chi^*(G)$). Let $\Delta(G)$ be the maximum degree of $G$ and let $$\Gamma(G)=\max \Big\{\frac{2|E(U)|}{|U|-1}:\,\, U \subseteq V, \,\, |U|\ge 3 \hskip 2mm {\rm and \hskip 2mm odd} \Big\},$$ where $E(U)$ is the set of all edges of $G$ with both ends in $U$. Clearly, $\max\{\Delta(G), \, \lceil \Gamma(G) \rceil \}$ is a lower bound for $\chi'(G)$. As shown by Seymour, $\chi^*(G)=\max\{\Delta(G), \, \Gamma(G)\}$. In the 1970s Goldberg and Seymour independently conjectured that $\chi'(G) \le \max\{\Delta(G)+1, \, \lceil \Gamma(G) \rceil\}$. Over the past four decades this conjecture, a cornerstone in modern edge-coloring, has been a subject of extensive research, and has stimulated a significant body of work. In this paper we present a proof of this conjecture. Our result implies that, first, there are only two possible values for $\chi'(G)$, so an analogue to Vizing's theorem on edge-colorings of simple graphs, a fundamental result in graph theory, holds for multigraphs; second, although it is $NP$-hard in general to determine $\chi'(G)$, we can approximate it within one of its true value, and find it exactly in polynomial time when $\Gamma(G)>\Delta(G)$; third, every multigraph $G$ satisfies $\chi'(G)-\chi^*(G) \le 1$, so FECP has a fascinating integer rounding property

Topological superfluid in a fermionic bilayer optical lattice

In this paper, a topological superfluid phase with Chern number C=1 possessing gapless edge states and non-Abelian anyons is designed in a C=1 topological insulator proximity to an s-wave superfluid on an optical lattice with the effective gauge field and layer-dependent Zeeman field coupled to ultracold fermionic atoms pseudo spin. We also study its topological properties and calculate the phase stiffness by using the random-phase-approximation approach. Finally we derive the temperature of the Kosterlitz-Thouless transition by means of renormalized group theory. Owning to the existence of non-Abelian anyons, this C=1 topological superfluid may be a possible candidate for topological quantum computation.Comment: 15 pages, 8 figure

GCS overexpression is associated with multidrug resistance of human HCT-8 colon cancer cells

<p>Abstract</p> <p>Purpose</p> <p>Multidrug resistance is one of the main impediments to the successful treatment of colon cancer. Glucosylceramide synthase (GCS) which is related to multidrug resistance (MDR) can reduce the level of ceramide and can help cells escape from the ceramide-induced cell apoptosis. However, the underlying mechanism is still unclear.</p> <p>Methods</p> <p>The cell proliferation and cell toxicity were measured with Cell Counting Kit-8 (CCK-8). The mRNA levels of GCS and MDR1 were detected by semiquantitative reverse transcription-PCR amplification, the protein levels of GCS, caspase-3 and P-gp proteins were indicated by Western blotting. The apoptosis rates of cells were measured with flow cytometry.</p> <p>Results</p> <p>The relative mRNA levels of GCS in HCT-8, HCT-8/VCR, HCT-8/VCR- sh-mock and HCT-8/VCR-sh-GCS were 71.4 ± 1.1%, 95.1 ± 1.2%, 98.2 ± 1.5%, and 66.6 ± 2.1% respectively. The mRNA levels of MDR1 were respectively 61.3 ± 1.1%, 90.5 ± 1.4%, 97.6 ± 2.2% and 56.1 ± 1.2%. The IC50 of Cisplatin complexes were respectively 69.070 ± 0.253 μg/ml, 312.050 ± 1.46 μg/ml, 328.741 ± 5.648 μg/ml, 150.792 ± 0.967 μg/ml in HCT-8, HCT-8/VCR, HCT-8/VCR-sh-mock and HCT-8/VCR-sh-GCS. The protein levels of caspase-3 were 34.2 ± o.6%, 93.0 ± 0.7%, 109.09 ± 0.7%, 42.7 ± 1.3% respectively. The apoptosis rates of cells were 8.77 ± 0.14%, 12.75 ± 0.54%, 15.39 ± 0.41% and 8.49 ± 0.23% respectively.</p> <p>Conclusion</p> <p>In conclusion, our research indicated that suppression of GCS restores the sensitivity of multidrug resistance colon cancer cells to drug treatment.</p