The set chromatic number of a graph

Abstract

For a nontrivial connected graph G, let c : V (G) → N be a vertex coloring of G where adjacent vertices may be colored the same. For a vertex v of G, the neighborhood color set NC(v) is the set of colors of the neighbors of v. The coloring c is called a set coloring if NC(u) = NC(v) for every pair u, v of adjacent vertices of G. The minimum number of colors required of such a coloring is called the set chromatic number χ s (G) of G. The set chromatic numbers of some well-known classes of graphs are determined and several bounds are established for the set chromatic number of a graph in terms of other graphical parameters.

Set vertex colorings and joins of graphs

“…In [1] the set chromatic numbers of some well-known graphs (namely cycles, bipartite graphs, and complete multipartite graphs) were determined. Furthermore, several bounds were established for the set chromatic number of a graph G in terms of other graphical parameters, namely the chromatic number χ(G) and the clique number ω(G), which is the order of a largest complete subgraph (clique) in G. Some of these results are stated below.…”

“…Let G and G ′ be the two copies of G in G + G. Note that χ s (G + G) χ s (G) + χ s (G ′ ) + 1 = 2k + 1 by Theorem 4. 1. To show that χ s (G + G ′ ) 2k + 1, assume, to the contrary, that χ s (G + G ′ ) = l 2k and let c : V (G + G ′ ) → N l be a set l-coloring of G + G ′ .…”

“…If G is a connected graph of order n, then χ s (G) = 1 if and only if χ(G) = 1 (in which case G = K 1 ) and χ s (G) = n if and only if χ(G) = n (in which case G = K n ). It was shown in [1] that χ s (G) = n − 1 if and only if χ(G) = n − 1 and that for each pair k, n of integers with 2 k n, there is a connected graph G of order n with χ s (G) = k. The following observation will be useful to us.…”

“…Certainly the most common graph colorings used to distinguish every two adjacent vertices in a graph G are the proper colorings, where distinct colors are assigned to every two adjacent vertices of G. The minimum number of colors required in a proper coloring of G is the chromatic number χ(G). In [1] another vertex coloring of graphs for the purpose of distinguishing every two adjacent vertices of G which may require fewer than χ(G) colors was introduced.…”

“…The neighborhood color set NC c (v) = c(N (v)) is the set of colors of the neighbors of v. (If the coloring c under consideration is clear, we write NC(v) for the neighborhood color set of v.) The coloring c is called set neighbor-distinguishing or simply a set coloring if NC(u) ̸ = NC(v) for every pair u, v of adjacent vertices of G. The minimum number of colors required of such a coloring is called the set chromatic number of G and is denoted by χ s (G). This concept was introduced and studied in [1] where it was observed that 1 χ s (G) χ(G) n for every graph G of order n. To illustrate these concepts, we consider the graph G of Fig. 1.…”