Ebrahim Salehi

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.

F-Continuous Graphs

For a nontrivial connected graph F, the F-degree of a vertex υ in a graph G is the number of copies of F in G containing υ. A graph G is F-continuous (or F-degree continuous) if the F-degrees of every two adjacent vertices of G differ by at most 1. All P3-continuous graphs are determined. It is observed that if G is a nontrivial connected graph that is F-continuous for all nontrivial connected graphs F, then either G is regular or G is a path. In the case of a 2-connected graph F, however, there always exists a regular graph that is not F-continuous. It is also shown that for every graph H and every 2-connected graph F, there exists an F-continuous graph G containing H as an induced subgraph.