\(J\)-coloring of graph operations. (English) Zbl 1430.05042
Summary: A vertex \(v\) of a given graph is said to be in a rainbow neighbourhood of \(G\) if every color class of \(G\) consists of at least one vertex from the closed neighbourhood \(N[v]\). A maximal proper coloring of a graph \(G\) is a \(J\)-coloring if and only if every vertex of \(G\) belongs to a rainbow neighbourhood of \(G\). In general all graphs need not have a \(J\)-coloring, even though they admit a chromatic coloring. In this paper, we characterise graphs which admit a \(J\)-coloring. We also discuss some preliminary results in respect of certain graph operations which admit a \(J\)-coloring under certain conditions.
MSC:
| 05C15 | Coloring of graphs and hypergraphs |
| 05C38 | Paths and cycles |
| 05C75 | Structural characterization of families of graphs |
| 05C85 | Graph algorithms (graph-theoretic aspects) |
| 05C76 | Graph operations (line graphs, products, etc.) |
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