A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc(G) of the graph G is the smallest num...A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc(G) of the graph G is the smallest number of colors in a linear coloring of G.In this paper, we prove that every graph G with girth g(G) and maximum degree Δ(G) that can be embedded in a surface of nonnegative characteristic has $ lc(G) = \left\lceil {\frac{{\Delta (G)}} {2}} \right\rceil + 1 $ if there is a pair (Δ, g) ∈ {(13, 7), (9, 8), (7, 9), (5, 10), (3, 13)} such that G satisfies Δ(G) ? Δ and g(G) ? g.展开更多
A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc（G） of the graph G is the smallest n...A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc（G） of the graph G is the smallest number of colors in a linear coloring of G. In this paper, it is proved that every planar graph G with girth g and maximum degree A has （1） lc（G） ≤ △ ＋ 21 if △ ≥ 9; （2） lc（G） ≤[△/2]＋ 7 if g≥5; （3） lc（G） ≤ [△/2]＋2ifg≥7and△ ≥7.展开更多
A linear coloring of a graph G is a proper vertex coloring such that the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc（G） of G is the sm...A linear coloring of a graph G is a proper vertex coloring such that the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc（G） of G is the smallest number of colors in a linear coloring of G. In this paper, we prove that every planar graph G with maximum degree 5 is 11-linear-colorable.展开更多
This paper investigates the selective liquid response for Morpho didius butterfly wing scales and propose an optical model to explain the effect of different components on the liquid response. It is found out that the...This paper investigates the selective liquid response for Morpho didius butterfly wing scales and propose an optical model to explain the effect of different components on the liquid response. It is found out that the reason of the selective response is that the liquid media forms nanometre-thick films between ridge-lamellae nanostructures and changes the constructive interference wavelength. There is linear relation between the structural color of ridge-lamellae structure and index of liquid background media. The reason of vapor＇s responses is that the nanometre-thick liquid fi lms on ridge-lamellae nanostructures change the constructive interference wavelength. These liquid films are formed due to vapor adsorption. Therefore,the selective linear liquid response can be applied to design nano-engineered photonic liquid and vapor sensors.展开更多
基金supported by National Natural Science Foundation of China (Grant No. 10771197)the Natural Science Foundation of Zhejiang Province of China (Grant No. Y607467)
文摘A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc(G) of the graph G is the smallest number of colors in a linear coloring of G.In this paper, we prove that every graph G with girth g(G) and maximum degree Δ(G) that can be embedded in a surface of nonnegative characteristic has $ lc(G) = \left\lceil {\frac{{\Delta (G)}} {2}} \right\rceil + 1 $ if there is a pair (Δ, g) ∈ {(13, 7), (9, 8), (7, 9), (5, 10), (3, 13)} such that G satisfies Δ(G) ? Δ and g(G) ? g.
基金Supported by National Natural Science Foundation of China (Grant Nos. 61070230, 10971121 and 61103199)NSFSP of China (Grant No. ZR2009AM009)
文摘A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc（G） of the graph G is the smallest number of colors in a linear coloring of G. In this paper, it is proved that every planar graph G with girth g and maximum degree A has （1） lc（G） ≤ △ ＋ 21 if △ ≥ 9; （2） lc（G） ≤[△/2]＋ 7 if g≥5; （3） lc（G） ≤ [△/2]＋2ifg≥7and△ ≥7.
基金Supported by the National Natural Science Foundation of China (Grant No. 11071223)the Natural ScienceFoundation of Zhejiang Province (Grant No. Z6090150)Research Project of Zhejiang Educational Committee(Grant No. Y201121311)
文摘A linear coloring of a graph G is a proper vertex coloring such that the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc（G） of G is the smallest number of colors in a linear coloring of G. In this paper, we prove that every planar graph G with maximum degree 5 is 11-linear-colorable.
基金Supported by the National Natural Science Foundation of China(51305129)the Natural Science Foundation of Hubei Province(Q20151411)
文摘This paper investigates the selective liquid response for Morpho didius butterfly wing scales and propose an optical model to explain the effect of different components on the liquid response. It is found out that the reason of the selective response is that the liquid media forms nanometre-thick films between ridge-lamellae nanostructures and changes the constructive interference wavelength. There is linear relation between the structural color of ridge-lamellae structure and index of liquid background media. The reason of vapor＇s responses is that the nanometre-thick liquid fi lms on ridge-lamellae nanostructures change the constructive interference wavelength. These liquid films are formed due to vapor adsorption. Therefore,the selective linear liquid response can be applied to design nano-engineered photonic liquid and vapor sensors.
