The minimum number of colors needed to properly color the vertices and edges of a graph G is called the total chromatic number of G and denoted by χ'' (G). It is shown that if a planar graph G has maximum deg...The minimum number of colors needed to properly color the vertices and edges of a graph G is called the total chromatic number of G and denoted by χ'' (G). It is shown that if a planar graph G has maximum degree Δ≥9, then χ'' (G) = Δ + 1. In this paper, we prove that if G is a planar graph with maximum degree 8 and without intersecting chordal 4-cycles, then χ ''(G) = 9.展开更多
Let G=(V,E)be a graph andφbe a total coloring of G by using the color set{1,2,...,k}.Let f(v)denote the sum of the color of the vertex v and the colors of all incident edges of v.We say thatφis neighbor sum distingu...Let G=(V,E)be a graph andφbe a total coloring of G by using the color set{1,2,...,k}.Let f(v)denote the sum of the color of the vertex v and the colors of all incident edges of v.We say thatφis neighbor sum distinguishing if for each edge uv∈E(G),f(u)=f(v).The smallest number k is called the neighbor sum distinguishing total chromatic number,denoted byχ′′nsd(G).Pil′sniak and Wo′zniak conjectured that for any graph G with at least two vertices,χ′′nsd(G)(G)+3.In this paper,by using the famous Combinatorial Nullstellensatz,we show thatχ′′nsd(G)2(G)+col(G)-1,where col(G)is the coloring number of G.Moreover,we prove this assertion in its list version.展开更多
Inorganic nanoparticles have been introduced into biological systems as useful probes for in vitro diagnosis and in vivo imaging, due to their relatively small size and exceptional physical and chemical properties. A ...Inorganic nanoparticles have been introduced into biological systems as useful probes for in vitro diagnosis and in vivo imaging, due to their relatively small size and exceptional physical and chemical properties. A new kind of color- tunable Gd-Zn-Cu-In-S/ZnS （GZCIS/ZnS） quantum dots （QDs） with stable crystal structure has been successfully synthesized and utilized for magnetic resonance （MR） and fluorescence dual modality imaging. This strategy allows successful fabrication of GZCIS/ZnS QDs by incorporating Gd into ZCIS/ZnS QDs to achieve great MR enhancement without compromising the fluorescence properties of the initial ZCIS/ZnS QDs. The as-prepared GZCIS/ZnS QDs show high T1 MR contrast as well as ＂color-tunable＂ photoluminescence （PL） in the range of 550-725 nm by adjusting the Zn/Cu feeding ratio with high PL quantum yield （QY）. The GZCIS/ZnS QDs were transferred into water via a bovine serum albumin （BSA） coating strategy. The resulting Cd-free GZCIS/ZnS QDs reveal negligible cytotoxicity on both HeLa and A549 cells. Both fluorescence and MR imaging studies were successfully performed in vitro and in vivo. The results demonstrated that GZCIS/ZnS QDs could be a dual-modal contrast agent to simultaneously produce strong MR contrast enhancement as well as fluorescence emission for in vivo imaging.展开更多
基金supported by Natural Science Foundation of Shandong Province (Grant No. ZR2009AM009)Scientific Research Foundation for the Excellent Middle-Aged and Youth Scientists of Shandong Province (Grant No. BS2012SF016)National Natural Science Foundation of China (Grant Nos.11001055 and 11101243)
文摘The minimum number of colors needed to properly color the vertices and edges of a graph G is called the total chromatic number of G and denoted by χ'' (G). It is shown that if a planar graph G has maximum degree Δ≥9, then χ'' (G) = Δ + 1. In this paper, we prove that if G is a planar graph with maximum degree 8 and without intersecting chordal 4-cycles, then χ ''(G) = 9.
基金supported by National Natural Science Foundation of China(Grant Nos.11101243 and 11371355)the Research Fund for the Doctoral Program of Higher Education of China(Grant No.20100131120017)the Scientific Research Foundation for the Excellent Middle Aged and Youth Scientists of Shandong Province of China(Grant No.BS2012SF016)
文摘Let G=(V,E)be a graph andφbe a total coloring of G by using the color set{1,2,...,k}.Let f(v)denote the sum of the color of the vertex v and the colors of all incident edges of v.We say thatφis neighbor sum distinguishing if for each edge uv∈E(G),f(u)=f(v).The smallest number k is called the neighbor sum distinguishing total chromatic number,denoted byχ′′nsd(G).Pil′sniak and Wo′zniak conjectured that for any graph G with at least two vertices,χ′′nsd(G)(G)+3.In this paper,by using the famous Combinatorial Nullstellensatz,we show thatχ′′nsd(G)2(G)+col(G)-1,where col(G)is the coloring number of G.Moreover,we prove this assertion in its list version.
文摘Inorganic nanoparticles have been introduced into biological systems as useful probes for in vitro diagnosis and in vivo imaging, due to their relatively small size and exceptional physical and chemical properties. A new kind of color- tunable Gd-Zn-Cu-In-S/ZnS （GZCIS/ZnS） quantum dots （QDs） with stable crystal structure has been successfully synthesized and utilized for magnetic resonance （MR） and fluorescence dual modality imaging. This strategy allows successful fabrication of GZCIS/ZnS QDs by incorporating Gd into ZCIS/ZnS QDs to achieve great MR enhancement without compromising the fluorescence properties of the initial ZCIS/ZnS QDs. The as-prepared GZCIS/ZnS QDs show high T1 MR contrast as well as ＂color-tunable＂ photoluminescence （PL） in the range of 550-725 nm by adjusting the Zn/Cu feeding ratio with high PL quantum yield （QY）. The GZCIS/ZnS QDs were transferred into water via a bovine serum albumin （BSA） coating strategy. The resulting Cd-free GZCIS/ZnS QDs reveal negligible cytotoxicity on both HeLa and A549 cells. Both fluorescence and MR imaging studies were successfully performed in vitro and in vivo. The results demonstrated that GZCIS/ZnS QDs could be a dual-modal contrast agent to simultaneously produce strong MR contrast enhancement as well as fluorescence emission for in vivo imaging.
