Distinguishing Chromatic Number of Cartesian Products of Graphs Hemanshu Kaul hkaul@math.uiuc.edu www.math.uiuc.edu/ ∼ hkaul/ . University of Illinois at Urbana-Champaign Graph Packing – p.1/14
Cartesian Product of Graphs Let G = ( V ( G ) , E ( G )) and H = ( V ( H ) , E ( H )) be two graphs. G ✷ H denotes the Cartesian product of G and H . V ( G ✷ H ) = { ( u, v ) | u ∈ V ( G ) , v ∈ V ( H ) } . vertex ( u, v ) is adjacent to vertex ( w, z ) if either u = w and vz ∈ E ( H ) or v = z and uw ∈ E ( G ) . Extend this definition to G 1 ✷ G 2 ✷ . . . ✷ G d . Denote G d = ✷ d i =1 G . Graph Packing – p.2/14
Cartesian Product of Graphs G H G H Graph Packing – p.2/14
Cartesian Product of Graphs A graph G is said to be a prime graph if whenever G = G 1 ✷ G 2 , then either G 1 or G 2 is a singleton vertex. Prime Decomposition Theorem [Sabidussi(1960) and Vizing(1963)] Let G be a connected graph, then G ∼ = G p 1 1 ✷ G p 2 2 ✷ . . . ✷ G p d d , where G i and G j are distinct prime graphs for i � = j , and p i are constants. Theorem [Imrich(1969) and Miller(1970)] All automorphisms of a cartesian product of graphs are induced by the automorphisms of the factors and by transpositions of isomorphic factors. Graph Packing – p.2/14
Chromatic Number Let G = ( V ( G ) , E ( G )) be a graph. Denote n ( G ) = | V ( G ) | , number of vertices in G . A proper k -coloring of G is a labeling of V ( G ) with k labels such that adjacent vertices get distinct labels. Chromatic Number, χ ( G ) , is the least k such that G has a proper k -coloring. Graph Packing – p.3/14
Chromatic Number Fact: Let G = ✷ d i =1 G i . Then χ ( G ) = max i =1 ,...,d { χ ( G i ) } Let f i be an optimal proper coloring of G i , i = 1 , . . . , d . f d : V ( G ) → { 0 , 1 , . . . , t − 1 } as Canonical Coloring d f d ( v 1 , v 2 , . . . , v d ) = � f i ( v i ) mod t , t = max i { χ ( G i ) } i =1 Graph Packing – p.3/14
Distinguishing Number A distinguishing k -labeling of G is a labeling of V ( G ) with k labels such that the only color-preserving automorphism of G is the identity. Distinguishing Number, D ( G ) , is the least k such that G has a distinguishing k -labeling. Introduced by Albertson and Collins in 1996. Since then, especially in the last five years, a whole class of research literature combining graphs and group actions has arisen around this topic. Graph Packing – p.4/14
Distinguishing Number Some motivating results : Theorem [Bogstad + Cowen, 2004] D ( Q d ) = 2 , for d ≥ 4 , where Q d is the d -dimensional hypercube. Theorem [Albertson, 2004] D ( G 4 ) = 2 , if G is a prime graph. Theorem [Klavzar + Zhu , 2005+] D ( G d ) = 2 , for d ≥ 3 . Follows from D ( K d t ) = 2 , for d ≥ 3 . Graph Packing – p.4/14
Distinguishing Chromatic Number A distinguishing proper k -coloring of G is a proper k -coloring of G such that the only color-preserving automorphism of G is the identity. Distinguishing Chromatic Number, χ D ( G ) , is the least k such that G has a distinguishing proper k -coloring. Graph Packing – p.5/14
Distinguishing Chromatic Number A distinguishing proper k -coloring of G is a proper k -coloring of G such that the only color-preserving automorphism of G is the identity. Distinguishing Chromatic Number, χ D ( G ) , is the least k such that G has a distinguishing proper k -coloring. A proper coloring of G that breaks all its symmetries. A proper coloring of G that uniquely determines the vertices. Graph Packing – p.5/14
� � � � Examples � � � � Not Distinguishing Graph Packing – p.6/14
� � � � Examples � � � � Not Distinguishing Graph Packing – p.6/14
Examples � � � � Distinguishing χ D ( P 2 n +1 ) = 3 and χ D ( P 2 n ) = 2 Graph Packing – p.6/14
Examples � � � � Distinguishing χ D ( P 2 n +1 ) = 3 and χ D ( P 2 n ) = 2 Not Distinguishing Graph Packing – p.6/14
Examples � � � � Distinguishing χ D ( P 2 n +1 ) = 3 and χ D ( P 2 n ) = 2 Distinguishing χ D ( C n ) = 3 except χ D ( C 4 ) = χ D ( C 6 ) = 4 Graph Packing – p.6/14
Motivation Distinguishing Chromatic Number, χ D ( G ) , is the least k such that G has a distinguishing proper k -coloring. The chromatic number, χ ( G ) , is an immediate lower bound for χ D ( G ) . How many more colors than χ ( G ) does χ D ( G ) need? Theorem [Collins + Trenk, 2006] χ D ( G ) = n ( G ) ⇔ G is a complete multipartite graph. χ D ( K n 1 ,n 2 ,...,n t ) = � t while χ ( K n 1 ,n 2 ,...,n t ) = t i =1 n i Graph Packing – p.7/14
Motivation Distinguishing Chromatic Number, χ D ( G ) , is the least k such that G has a distinguishing proper k -coloring. The chromatic number, χ ( G ) , is an immediate lower bound for χ D ( G ) . How many more colors than χ ( G ) does χ D ( G ) need? Theorem [Collins + Trenk, 2006] χ D ( G ) = n ( G ) ⇔ G is a complete multipartite graph. χ D ( K n 1 ,n 2 ,...,n t ) = � t while χ ( K n 1 ,n 2 ,...,n t ) = t i =1 n i Theorem [Collins + Trenk, 2006] χ D ( G ) ≤ 2∆( G ) , with equality iff G = K ∆ , ∆ or C 6 . Graph Packing – p.7/14
Main Theorem Theorem 1 [Choi + Hartke + K., 2005+] Let G be a graph. Then there exists an integer d G such that for all d ≥ d G , χ D ( G d ) ≤ χ ( G ) + 1 . By the Prime Decomposition Theorem for Graphs, G = G p 1 1 ✷ G p 2 2 ✷ . . . ✷ G p k k , where G i are distinct prime graphs. i =1 ,...,k { lg n ( G i ) d G = max } + 5 Then, p i Note, n ( G ) = ( n ( G 1 )) p 1 ∗ ( n ( G 2 )) p 2 ∗ · · · ∗ ( n ( G k )) p k At worst, d G = lg n ( G ) + 5 Graph Packing – p.8/14
Main Theorem Theorem 1 [Choi + Hartke + K., 2005+] Let G be a graph. Then there exists an integer d G such that for all d ≥ d G , χ D ( G d ) ≤ χ ( G ) + 1 . i =1 ,...,k { lg n ( G i ) d G = max } + 5 p i when, n ( G ) = ( n ( G 1 )) p 1 ∗ ( n ( G 2 )) p 2 ∗ · · · ∗ ( n ( G k )) p k d G is unlikely to be a constant, as the example of Complete Multipartite Graphs indicates − pushing χ D ( K n 1 ,n 2 ,...,n t ) down from n ( G ) to t + 1 ! Graph Packing – p.8/14
Proof Idea for Theorem 1 Fix an optimal proper coloring of G . Embed G in a complete multipartite graph H . Form H by adding all the missing edges between the color classes of G . Now work with H . BUT G ⊆ H � χ D ( G ) ≤ χ D ( H ) ! Graph Packing – p.9/14
Proof Idea for Theorem 1 Fix an optimal proper coloring of G . Embed G in a complete multipartite graph H . Form H by adding all the missing edges between the color classes of G . Then construct a distinguishing proper coloring of H d that is also a distinguishing proper coloring of G d . Study Distinguishing Chromatic Number of Cartesian Products of Complete Multipartite Graphs. Graph Packing – p.9/14
Hamming Graphs and Hypercubes Theorem 2 [Choi + Hartke + K., 2005+] χ D ( ✷ d Given t i ≥ 2 , i =1 K t i ) ≤ max i { t i } + 1 , for d ≥ 5 . Graph Packing – p.10/14
Hamming Graphs and Hypercubes Theorem 2 [Choi + Hartke + K., 2005+] χ D ( ✷ d Given t i ≥ 2 , i =1 K t i ) ≤ max i { t i } + 1 , for d ≥ 5 . χ D ( K d Corollary : Given t ≥ 2 , t ) ≤ t + 1 , for d ≥ 5 . Both these upper bounds are 1 more than their respective lower bounds. Graph Packing – p.10/14
Hamming Graphs and Hypercubes Theorem 2 [Choi + Hartke + K., 2005+] χ D ( ✷ d Given t i ≥ 2 , i =1 K t i ) ≤ max i { t i } + 1 , for d ≥ 5 . χ D ( K d Corollary : Given t ≥ 2 , t ) ≤ t + 1 , for d ≥ 5 . Both these upper bounds are 1 more than their respective lower bounds. Corollary : χ D ( Q d ) = 3 , for d ≥ 5 . Graph Packing – p.10/14
Complete Multipartite Graphs Theorem 3 [Choi + Hartke + K., 2005+] Let H be a complete multipartite graph. Then χ D ( H d ) ≤ χ ( H ) + 1 , for d ≥ lg n ( H ) + 5 . This is already enough to prove Theorem 1 for prime graphs. Graph Packing – p.11/14
Complete Multipartite Graphs Theorem 3 [Choi + Hartke + K., 2005+] Let H be a complete multipartite graph. Then χ D ( H d ) ≤ χ ( H ) + 1 , for d ≥ lg n ( H ) + 5 . This is already enough to prove Theorem 1 for prime graphs. Theorem 4 [Choi + Hartke + K., 2005+] i =1 H p i Let H = ✷ k i , where H i are distinct complete multipartite graphs. Then χ D ( H d ) ≤ χ ( H ) + 1 , i =1 ,...,k { lg n i for d ≥ max p i } + 5 , where n i = n ( H i ) . Graph Packing – p.11/14
Main Theorem Theorem 1 [Choi + Hartke + K., 2005+] Let G be a graph. Then there exists an integer d G such that for all d ≥ d G , χ D ( G d ) ≤ χ ( G ) + 1 . By the Prime Decomposition Theorem for Graphs, G = G p 1 1 ✷ G p 2 2 ✷ . . . ✷ G p k k , where G i are distinct prime graphs. i =1 ,...,k { lg n ( G i ) d G = max } + 6 Then, p i Graph Packing – p.12/14
Outline of the Proof for Hamming Graphs Start with the canonical proper coloring f d of cartesian products of graphs, f d : V ( K d t ) → { 0 , 1 , . . . , t − 1 } with d f d ( v ) = f ( v i ) mod t , � i =1 where f ( v i ) = i is an optimal proper coloring of K t . Graph Packing – p.13/14
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