DISTINGUISHING CHROMATIC NUMBER OF CARTESIAN PRODUCTS OF GRAPHS

作者：Choi Jeong Ok^*; Hartke Stephen G; Kaul Hemanshu

来源：SIAM Journal on Discrete Mathematics, 2010, 24(1): 82-100.

DOI：10.1137/060651392

• symmetry breaking
• graph automorphism
• distinguishing number
• distinguishing chromatic number
• Cartesian product of graphs
• graph coloring

The distinguishing chromatic number chi(D) (G) of a graph G is the least integer k such that there is a proper k-coloring of G which is not preserved by any nontrivial automorphism of G. We study the distinguishing chromatic number of Cartesian products of graphs by focusing on how much it can exceed the trivial lower bound of the chromatic number chi(.). Our main result is that for every graph G, there exists a constant d(G) such that for all d >= d(G) the distinguishing chromatic number of G(d) is at most chi(G) + 1, where G(d) is the Cartesian product of d copies of G. We also prove that for d >= 5, the Cartesian product of d complete graphs has distinguishing chromatic number at most one more than the corresponding chromatic number, and we determine the distinguishing chromatic number of hypercubes exactly.

• 出版日期2010