Equitable list coloring of planar graphs without 4- and 6-cycles

AbstractA graph G is equitably k-choosable if for any k-uniform list assignment L, there exists an L-colorable of G such that each color appears on at most ⌈|V(G)|k⌉ vertices. Kostochka, Pelsmajer and West introduced this notion and conjectured that G is equitably k-choosable for k>Δ(G). We prove this for planar graphs with Δ(G)≥6 and no 4- or 6-cycles

Acta Mathematica Scientia 2006,26B(3):477-482 EDGE-FACE CHROMATIC NUMBER OF 2-CONNECTED PLANE GRAPHS WITH HIGH MAXIMUM DEGREE ∗

Abstract The edge-face chromatic number χef(G) of a plane graph G is the least number of colors assigned to the edges and faces such that every adjacent or incident pair of them receives different colors. In this article, the authors prove that every 2-connected plane graph G with ∆(G) ≥ |G | − 2 ≥ 9 has χef(G) = ∆(G). Key words Plane graph, edge-face chromatic number, edge chromatic number, maxi-mum degree 2000 MR Subject Classification 05C1

Adjacent vertex distinguishing edge-colorings of planar graphs with girth at least six

An adjacent vertex distinguishing edge-coloring of a graph G is a proper edge-coloring o G such that any pair of adjacent vertices are incident to distinct sets of colors. The minimum number of colors required for an adjacent vertex distinguishing edge-coloring of G is denoted by χ'ₐ(G). We prove that χ'ₐ(G) is at most the maximum degree plus 2 if G is a planar graph without isolated edges whose girth is at least 6. This gives new evidence to a conjecture proposed in [Z. Zhang, L. Liu, and J. Wang, Adjacent strong edge coloring of graphs, Appl. Math. Lett., 15 (2002) 623-626.

Fundamental Flotation Behaviors of Chalcopyrite and Galena Using O-Isopropyl-N-Ethyl Thionocarbamate as a Collector

Copper and lead are two important and widely used metals in industry. Chalcopyrite (CuFeS2) is associated with galena (PbS) in ore, and it has been a research hotspot in separating galena from chalcopyrite by flotation. In this study, the flotation behaviors of chalcopyrite and galena were studied through flotation tests, adsorption measurements, solution chemistry calculation, Fourier transform infrared spectroscopy (FTIR) and molecular dynamics (MD) simulations. The results show that the floatability of chalcopyrite is better than that of galena in the presence of O-isopropyl-N-ethyl thionocarbamate (IPETC), and the recovery difference between chalcopyrite and galena is about 20% when IPETC is 7 × 10−4 mol/L at pH 9.5, while the floatability difference between the two minerals is significant. Competitive adsorption of OH− and IPETC on mineral surfaces leads to lower floatability of galena than that of chalcopyrite. IPETC is able to remove the hydration layer on mineral surfaces and then adsorb on active sites. The floatability of minerals is enhanced with the increase of their hydrophobicity. This study provides a reference to separate galena from chalcopyrite