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Copyright © by SIAM. Unauthorized Reproduction of This Article Is Prohibited SIAM J. DISCRETE MATH. c 2010 Society for Industrial and Applied Mathematics Vol. 24, No. 1, pp. 82–100 DISTINGUISHING CHROMATIC NUMBER OF CARTESIAN PRODUCTS OF GRAPHS ∗ † ‡ § JEONG OK CHOI , STEPHEN G. HARTKE , AND HEMANSHU KAUL Abstract. χ G G k The distinguishing chromatic number D ( ) of a graph is the least integer 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 χ · much it can exceed the trivial lower bound of the chromatic number ( ). Our main result is that G d d ≥ d for every graph , there exists a constant G such that for all G the distinguishing chromatic Gd χ G Gd d G number of is at most ( )+1, where is the Cartesian product of copies of .Wealsoprove 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. Key words. symmetry breaking, graph automorphism, distinguishing number, distinguishing chromatic number, Cartesian product of graphs, graph coloring AMS subject classifications. 05C25, 05C15 DOI. 10.1137/060651392 1. Introduction. Aproperk-coloring of a graph is a coloring of its vertices using at most k colors such that no two adjacent vertices get the same color. The chromatic number χ(G) of a graph G is the minimum k such that G has a proper k -coloring. Proper coloring of graphs is a well-studied, fundamental concept in graph theory (see [16]). In this paper, we study those proper colorings of a graph that break all its symmetries, and consequently, uniquely identify each of its vertices. More precisely, a coloring f : V (G) →{1,...,k} of a graph G is said to be a distinguishing proper k-coloring of G if it is a proper coloring of G and the identity automorphism is the only color-preserving automorphism of G.Thedistinguishing chromatic number χ G G k G k D ( )of is the minimum such that has a distinguishing proper -coloring. The distinguishing chromatic number was introduced by Collins and Trenk [10] as a natural specialization of the distinguishing number of a graph. A distinguish- ing k-labeling and the distinguishing number of a graph are defined analogously to distinguishing proper k-coloring and distinguishing chromatic number, respectively, without the restriction that the labeling be a propercoloring. Since its introduction by Albertson and Collins in [3], the distinguishing number of a graph has become an active area of research (see [2], [5], [7], [15], among others). A theme common to most results on the distinguishing number of graphs is that usually this parameter is the smallest value possible, 2 or 1 if a graph has only the trivial automorphism. That is, two labels are often sufficient to break all symmetries of a given graph. This indicates that the distinguishing number of a graph does not strongly depend on its structure. A more pertinent point of view considers labeling a set (instead of a graph) that breaks all the actions ofa group acting on it (instead of ∗Received by the editors January 30, 2006; accepted for publication (in revised form) August 23, 2009; published electronically February 19, 2010. http://www.siam.org/journals/sidma/24-1/65139.html †Department of Mathematics, University of Illinois, Urbana, IL 61801 ([email protected]). ‡Department of Mathematics, University of Nebraska, Lincoln, NE 68588-0130 (shartke2@math. unl.edu). §Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616 (kaul@ math.iit.edu). 82 Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. DISTINGUISHING CHROMATIC NUMBER 83 the automorphism group of the graph). This general concept captures the real essence of distinguishing and indicates why algebraic techniques are more useful than graph theoretic methods. See [17], [8], [19], [9], and [20] for some of the related results. Similar concepts of distinguishing vertices of a graph through colorings, such as vertex-distinguishing edge-colorings of graphs (see, e.g., [1], [6], and [4]), have also been considered by other researchers. Collins and Trenk [10] introduced distinguishing χ chromatic number D for some basic families of graphs. They also prove an analogue χ G ≤ G G of Brooks’ theorem for proper colorings by showing that D ( ) 2Δ( ), where Δ( ) is the maximum degree of G, and characterizing the corresponding extremal graphs. Since the chromatic number χ(G)ofG is a trivial lower bound for χD (G), we are χ G χ G interested in finding conditions under which D ( )iscloseto ( ). In this paper, we consider the distinguishing chromatic number of Cartesian prod- ucts of graphs. The Cartesian product of two graphs G and H, denoted by G2H, is the graph with vertex set V (G2H)={(u, v) | u ∈ V (G),v ∈ V (H)}.Thevertex (u, v) is adjacent to the vertex (w, z) if either u = w and vz ∈ E(H)orv = z and uw ∈ E G ( ). Since this graph product is clearly commutative and associative (see G 2G 2 ...2G Gd 2d G [12]), this definition can be extended to 1 2 d, in particular = i=1 . Note that the d-dimensional hypercube Qd is defined to be a Cartesian product of d 2 K χ G complete graphs i=1 2. Our results show that D ( ) can be at most one worse than χ(G)whenG is a large Cartesian product of graphs. In this paper, all graphs that we consider are simple (i.e., with no loops or multi- edges) and connected, unless indicated otherwise. The notation lg indicates logarithm with base 2, and Z+ the positive integers. Our first result shows that there is a distinguishing proper coloring of a Cartesian product of complete graphs using at most one color more than the corresponding chromatic number. Theorem. d ≥ t ≥ i ,...,d {t }≤χ 2d K ≤ Given 5 and i 2, =1 , max i D ( i=1 ti ) max {ti} +1. The bound for d in the above theorem is sharp. Also as a corollary to the above theorem we determine the distinguishing chromatic number of large hypercubes. Theorem. χ (Qd)=3,ford ≥ 5. D A complete graph is a particular instance of a complete multipartite graph. We extend the above theorem on complete graphs to Cartesian products of complete multipartite graphs. Theorem. H 2r Hpi H If = i=1 i ,where i are distinct connected complete multipar- p ∈ Z+ χ H ≤ χ 2d H ≤ χ H tite graphs with at least two vertices and i ,then ( ) D ( i=1 ) ( )+1 2lg ni d ≥ i ,...,r{ } ni Hi i ,...,r for max =1 pi +5,where are the orders of for =1 . As shown in the next section, χ(H)=maxi{χ(Hi )}. Note that the above theorem is unlikely to be true for a constant d. Collins and χ G G G Trenk showed in [10] that D ( ) equals the number of vertices in if and only if is a complete multipartite graph. Thus, the distinguishing chromatic number and the chromatic number of a complete multipartite graph can be arbitrarily far apart. By taking only a fixed number of products, we feel it is unlikely that the distinguishing chromatic number and the chromatic number of these graphs can be made almost equal; see Conjecture 22 for a precise statement. χ G χ G G Our main result shows that D ( ) can be at most one more than ( )when is a large Cartesian power of any graph. Theorem. Let G be a connected graph with at least two vertices. Then there d exists an integer dG such that for all d ≥ dG, χ(G) ≤ χ (G ) ≤ χ(G)+1. D Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 84 JEONG OK CHOI, STEPHEN G. HARTKE, AND HEMANSHU KAUL r pi + Moreover, if G = 2 G ,withGi distinct factors of G and pi ∈ Z ,then i=1 i 2lg ni dG =maxi=1,...,r{ p } +5suffices, where ni is the order of Gi for i =1,...,r. i p Note that a “prime factorization” of a graph of the form G = 2r G i can be i=1 i found in polynomial time for any connected graph; prime factors of a graph are defined 2lg ni dG i ,...,r{ } in the next section. The function =max=1 pi + 5 is minimized for this prime factorization. Our results can be seen as analogous to recent results of [5], [7], [2], and [15] on a distinguishing number of graphs. Bogstad and Cowen [5] showed that the hypercubes of dimension at least 4 have a distinguishing number equal to 2. Chan [7] proved further such results about some hypercube generalizations. Recently, Albertson [2] and Klavˇzar and Zhu [15] extended this result to show that two colors suffice to distinguish a large enough Cartesian power of a graph. Note that the distinguishing number of a graph is equal to 1 if and only if it has no nontrivial automorphisms. Thus, for most interesting classes of graphs, at least one extra color will be required for distinguishing the vertices, and this one extra color is shown to suffice for Cartesian products of various graphs. Analogously, for the distinguishing chromatic number we show that using just one extra color than the minimum possible χ(G) colors is enough to give a distinguishing proper coloring for large enough hypercubes, Cartesian products of complete graphs, and Cartesian powers of any graph. Our main method of proof is to uniquely identify the vertices of a graph by reconstructing a canonical vector representation (defined in terms of the Cartesian product) using only the colors of the vertices and the structure of the graph.
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