| 摘要: | A proper vertex coloring of a graph G is a partition \{A_1,A_2,\ldots ,A_k\} of the vertex set V(G) into stable sets. For a graph G with a positive vertex-weight c:V(G) \rightarrow (0,\infty ), denoted by (G,c), let \chi (G,c) be the minimum value of \sum _{i=1}^k \max _{v \in A_i} c(v) over all proper vertex coloring \{A_1,A_2,\ldots ,A_k\} of G and \sharp \chi (G,c) the minimum value of k for a proper vertex coloring \{A_1,A_2,\ldots ,A_k\} of G such that \sum _{i=1}^k \max _{v \in A_i} c(v) = \chi (G,c). This paper establishes an upper bound on \sharp \chi (G,c) for a weighted r-colorable graph (G,c), and a Nordhaus–Gaddum type inequality for \chi (G,c). It also studies the c-perfection for a weighted graph (G,c). |
