Futaba Okamoto

The set chromatic number of a graph

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) 6= 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 χs(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.

On multiset colorings of graphs

A vertex coloring of a graph G is a multiset coloring if the multisets of colors of the neighbors of every two adjacent vertices are different. The minimum k for which G has a multiset k-coloring is the multiset chromatic number χm(G) of G. For every graph G, χm(G) is bounded above by its chromatic number χ(G). The multiset chromatic numbers of regular graphs are investigated. It is shown that for every pair k, r of integers with 2 ≤ k ≤ r − 1, there exists an r-regular graph with multiset chromatic number k. It is also shown that for every positive integer N , there is an r-regular graph G such that χ(G)−χm(G) = N . In particular, it is shown that χm(Kn × K2) is asymptotically √ n. In fact, χm(Kn×K2) = χm(cor(Kn+1)). The corona cor(G) of a graph G is the graph obtained from G by adding, for each vertex v in G, a new vertex v and the edge vv. It is shown that χm(cor(G)) ≤ χm(G) for every nontrivial connected graph G. The multiset chromatic numbers of the corona of all complete graphs are determined. 138 F. Okamoto, E. Salehi and P. Zhang From this, it follows that for every positive integer N , there exists a graph G such that χm(G) − χm(cor(G)) ≥ N . The result obtained on the multiset chromatic number of the corona of complete graphs is then extended to the corona of all regular complete multipartite graphs.