Let G be a bipartite graph with vertex set V(G) and edge set E(G), and let g and f be two nonnegative integer-valued functions defined on V(G) such that g(x) ≤f(x)for every vertex x of V(G). A(g,f)-coloring of G is a generalized edge-coloring in which each color appears at each vertex x at least g(x) and at most f(x) times. In this paper a polynomial algorithm to find a(g, f)-coloring of a bipartite graph with some constraints using the minimum number of colors is given. Furthermore, we show that the results in this paper are best possible.
LIU Guizhen & DENG Xiaotie Department of Mathematics, Shandong University, Jinan 250100, China
Let G be a bipartite graph and g and f be two positive integer-valued functions defined on vertex set V(G) of G such that g(x)≤f(x).In this paper,some sufficient conditions related to the connectivity and edge-connectivity for a bipartite (mg,mf)-graph to have a (g,f)-factor with special properties are obtained and some previous results are generalized.Furthermore,the new results are proved to be the best possible.
A new heteronuclear complex, dimer [(2-OH2)2Ba2(H2O)4][VO2(tpa)]24H2O (tpa: 2,4,6-tripicolinate trianion), was synthesized and structurally determined by X-ray diffraction technique where all hydrogen atoms have been located directly. The coordination geometry of V(V) ion is a distorted trigonal bipyrimid and that of Ba(II) ion is a capped square antiprism. One 2-O bridging bond and two hydrogen bonds act between the coordination geometries. A 1-D extended porous construction containing 20-member cavities is observed in the crystal.
