The 2-distance coloring of the Cartesian product of cycles using - PowerPoint PPT Presentation

The 2-distance coloring of the Cartesian product of cycles using optimal Lee codes Jon-Lark Kim Department of Mathematics University of Louisville Joint work with Seog-Jin Kim Konkuk University, Korea 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing University of Louisville, KY May 12, 2011

Outline • Introduction • k-Distance Coloring and Codes over Z m . • Recent Results • Conclusion

Introduction • Let G = ( V , E ) be a (finite or infinite) graph. • A k -distance coloring of G is a vertex coloring of G such that any two distinct vertices at distance less than or equal to k are assigned different colors. • When k = 1, we have a usual (vertex) coloring. The k -distance chromatic number of a graph G is the minimum number of colors necessary to k -distance color G , which is denoted by χ k ( G ) . • Let G k be the k th graph in which its vertex set is V ( G ) and there is an edge between two vertices of G k if and only if they have distance at most k . • Hence χ k ( G ) is equal to χ ( G k ) .

Cartesian product of graphs • Let G 1 and G 2 be graphs. • We consider the usual Cartesian (or box) product G 1 � G 2 . • The vertex set of G 1 � G 2 is the Cartesian product V ( G 1 ) × V ( G 2 ) of V ( G 1 ) and V ( G 2 ) . • There is an edge between two vertices of G 1 � G 2 if and only if they are adjacent in exactly one coordinate and agree in the other.

2-distance coloring of hypercube Q n • The graph for the k -distance coloring related to binary codes is Hamming graph H ( 2 , n ) (or hypercube Q n = P 2 � P 2 � · · · � P 2 ).

2-distance coloring of hypercube Q n • The graph for the k -distance coloring related to binary codes is Hamming graph H ( 2 , n ) (or hypercube Q n = P 2 � P 2 � · · · � P 2 ). • The 2-distance coloring of Q n has been studied by Fertin-Godard-Raspaud (’03), Jamison-Matthews-Villalpando (’06), Kim-Du-Pardolos (’00), Ngo-Du-Graham (’00), ¨ Osterg˚ ard (’04). • Although Q n is a very simple graph, the exact 2-distance chromatic number of Q n is known only for some n . .R.J. ¨ • P Osterg˚ ard [On a hypercube coloring problem, J. Combin. Theory Ser. A, 108 (2004) 199–204]. χ 2 ( Q n ) χ 3 ( Q n ) lim n →∞ = 1 and lim n →∞ = 2. n n

2-distance coloring of C m 1 � C m 2 • The graph for the k -distance coloring related to non-binary codes is the Cartesian product cycles.

2-distance coloring of C m 1 � C m 2 • The graph for the k -distance coloring related to non-binary codes is the Cartesian product cycles. • Recently, Eric Sopena and Jiaojiao Wu [Coloring the square of the Cartesian product of two cycles, Discrete Math. 310 pp. 2327-2333, 2010] studied χ 2 ( C m 1 � C m 2 ) , where C m 1 and C m 2 are cycles of lengths m 1 and m 2 , respectively. • They proved χ 2 ( C m 1 � C m 2 ) ≤ 7 except when ( m 1 , m 2 ) = ( 3 , 3 ) in which case the value is 9, and when ( m 1 , m 2 ) = ( 4 , 4 ) or ( 5 , 5 ) , in which case the value is 8. • Also, A. Por and D.R. Wood [Colourings of the cartesian product of graphs and multiplicative Sidon sets, Combinatorica 29 (4) pp. 449-466, 2009] proved that the chromatic number of the square of the Cartesian product of d cycles is at most 6 d + O ( log d ) .

Connection with codes over Z n m • We consider n copies of m -cycles. • It was less known about the k -distance coloring on the Cartesian product of n copies of m -cycles, where k ≥ 2 and n ≥ 3. • In fact, this problem is closely related to finding optimal codes in Z n m with minimum distance k + 1 with respect to Lee metric. • Since k = 2 and n = 3 are the first unknown cases, we focus on χ 2 ( C m � C m � C m ) for a positive integer m ≥ 3.

Representation of G ( n , m ) as Z n m • Let G ( n , m ) := C m � C m � · · · � C m . • We may represent G ( n , m ) as Z n m , where Z m is the ring of integers modulo m . • The Lee weight wt L ( u ) of u = ( u 1 , . . . , u n ) ∈ Z n m is defined as wt L ( u ) = � n i = 1 min { u i , m − u i } . • The Lee distance d L ( u , v ) of u = ( u 1 , . . . , u n ) and u = ( v 1 , . . . , v n ) is wt L ( u − v ) . • Then the distance between u and v in G ( n , m ) is the same as the Lee distance of u and v in Z n m . • This shows that there is an edge between u and v in G ( n , m ) if and only if their Lee distance d L ( u , v ) is 1. • This correspondence connects coloring problems with coding theory problems with respect to the Lee weight.

Lower bounds for χ 2 ( G ( n , m )) As the degree of any vertex of G ( n , m ) is 2 n , we have a trivial lower bound below. Lemma 1 χ 2 ( G ( n , m )) ≥ 2 n + 1. The lower bound can be met. Theorem 1 There exists a perfect code C on the graph G ( n , m ) if and only if χ 2 ( G ( n , m )) = 2 n + 1.

Corollary 1 If 2 n + 1 divides m , then χ 2 ( G ( n , m )) = 2 n + 1. Proof Golomb and Welch [Perfect codes in the Lee metric and the packing of polyominoes, SIAM J. Appl. Math. 18 (1970) 302–317] showed that there is a perfect 1-error-correcting Lee code in Z n m if 2 n + 1 | m and conjectured that there is no perfect t -error-correcting Lee code in Z n m if n > 2 and t > 1. Thus the claim follows from Theorem 1. Corollary 2 If χ 2 ( G ( n , m )) = 2 n + 1, then 2 n + 1 divides m n . Proof If χ 2 ( G ( n , m )) = 2 n + 1, then there exists a perfect code C on G ( m , n ) by Theorem 1. Thus ( 2 n + 1 ) · | C | = m n , that is, 2 n + 1 divides m n , as required.

Sphere packing bound for codes in Z n m Lemma (the sphere packing bound) Let A L m ( n , d ) be the size of an optimal code in Z n m with Lee distance d . Then m n � � A L m ( n , d ) ≤ , V m ( n , e ) where e = ⌊ ( d − 1 ) / 2 ⌋ and V m ( n , e ) is the volume of the ball of radius e around any vertex of Z n m . In particular, if n = 3 and d = 3, then A L m ( 3 , 3 ) ≤ ⌊ m 3 / 7 ⌋ . Furthermore, if G = C m � C m � C m , then α ( G 2 ) = A L m ( 3 , 3 ) and ⌈ m 3 /α ( G 2 ) ⌉ ≤ χ ( G 2 ) = χ 2 ( G ) , where α ( G 2 ) is the size of maximum independent set in G 2 .

Theorem 2 χ 2 ( C 3 � C 3 � C 3 ) = 9 Proof Let G = C 3 � C 3 � C 3 . We show that α ( G 2 ) = 3. By the sphere packing bound, one has A L 3 ( 3 , 3 ) ≤ 3. On the other hand, C = { ( 0 , 0 , 0 ) , ( 1 , 1 , 1 ) , ( 2 , 2 , 2 ) } is a linear code with d ( C ) ≥ 3, implying A L 3 ( 3 , 3 ) ≥ 3. Hence α ( G 2 ) = A L 3 ( 3 , 3 ) = 3. Therefore χ 2 ( C 3 � C 3 � C 3 ) ≥ 9. Each ball of radius one around a codeword of C is colored with 7 colors. Then there are 6 vertices in G left uncolored. They form a set A ∪ B , where A = { ( 0 , 1 , 2 ) , ( 2 , 0 , 1 ) , ( 1 , 2 , 0 ) } and B = { ( 0 , 2 , 1 ) , ( 2 , 1 , 0 ) , ( 1 , 0 , 2 ) } . It is easy to check that A = ( 0 , 1 , 2 ) + C and B = ( 0 , 2 , 1 ) + C . Hence A and B respectively have distance 3. We color A with an 8th color and B with a 9th color. Thus χ 2 ( C 3 � C 3 � C 3 ) ≤ 9. Therefore we have χ 2 ( C 3 � C 3 � C 3 ) = 9.

Lemma 2 A L 4 ( 3 , 3 ) = 8 Proof Let C 4 = { ( 0 , 0 , 0 ) , ( 0 , 1 , 2 ) , ( 2 , 0 , 1 ) , ( 1 , 2 , 0 ) , ( 1 , 3 , 2 ) , ( 2 , 1 , 3 ) , ( 3 , 2 , 1 ) , ( 3 , 3 , 3 ) } . One can check d ( C 4 ) = 3. Note that C 4 is invariant under the cyclic shift. Hence A L 4 ( 3 , 3 ) ≥ 8. Now we will show that A L 4 ( 3 , 3 ) ≤ 8. Let C ⊂ Z 3 4 be a 4-ary code with d ( C ) = 3. Put R j := { y = ( y 1 , y 2 , y 3 ) ∈ Z 3 4 | y 1 = j } and C j := C ∩ R j for j = 0 , . . . , 3. Then for each j , | C j | ≤ 3 since A L 4 ( 2 , 3 ) ≤ ⌊ 16 5 ⌋ = 3 by the sphere packing bound. However we show that | C j | � = 3 for any j . Suppose that | C j | = 3 for some j . We may assume that ( j , 0 , 0 ) ∈ C j . Then we can check that C j ⊂ { ( j , 0 , 0 ) , ( j , 1 , 2 ) , ( j , 2 , 1 ) , ( j , 2 , 2 ) , ( j , 2 , 3 ) , ( j , 3 , 2 ) } . Since d ( C j ) = 3, we have | C j | ≤ 2. Therefore | C | ≤ � 3 j = 0 | C j | ≤ 4 · 2 = 8, completing the proof.

Theorem 3 χ 2 ( C 4 � C 4 � C 4 ) = 8 Proof Let G = C 4 � C 4 � C 4 . Then α ( G 2 ) = 8 by Lemma 2; hence χ 2 ( G ) ≥ 8. To show that the bound is tight, we use the code C 4 in Lemma 2. Just as in Theorem 2, each ball of radius one around a codeword of C 4 is colored with 7 colors. There are 64 − 7 · 8 = 8 vertices in G left uncolored. They form a set A := ( 2 , 2 , 2 ) + C . Hence d ( A ) = 3. We color A with an 8th color. Therefore χ 2 ( G ) = 8.

Theorem 4 χ 2 ( C 5 � C 5 � C 5 ) = 9 Proof Let G = C 5 � C 5 � C 5 . It is known that A L 5 ( 3 , 3 ) = 15 (see Quistorff, 2006). Hence α ( G 2 ) = 15; hence χ 2 ( G ) ≥ 9. Showing that χ 2 ( G ) ≤ 9 is technical, hence omitted.

A L 6 ( 3 , 3 ) = 26 What is the size of an optimal code in Z 3 6 with d = 3, that is, A L 6 ( 3 , 3 ) ?

A L 6 ( 3 , 3 ) = 26 What is the size of an optimal code in Z 3 6 with d = 3, that is, A L 6 ( 3 , 3 ) ? By the Sphere-Packing bound, A L 6 ( 3 , 3 ) ≤ 30 (see also Quistorff, 2006). For the first time, we show that A L 6 ( 3 , 3 ) = 26. Lemma 3 A L 6 ( 3 , 3 ) = 26

Example of | C | = 26 Its proof is nontrivial and omitted. We just give one C with size 26 and Lee distance 3. C = { ( 4 , 1 , 0 ) , ( 4 , 4 , 0 ) , ( 2 , 3 , 0 ) , ( 1 , 0 , 0 ) , ( 0 , 2 , 0 ) , ( 0 , 5 , 1 ) , ( 3 , 5 , 1 ) , ( 5 , 3 , 1 ) , ( 2 , 1 , 1 ) , ( 4 , 2 , 2 ) , ( 2 , 4 , 2 ) , ( 5 , 0 , 2 ) , ( 1 , 2 , 2 ) , ( 5 , 4 , 3 ) , ( 1 , 5 , 3 ) , ( 3 , 3 , 3 ) , ( 0 , 1 , 3 ) , ( 3 , 0 , 3 ) , ( 1 , 3 , 4 ) , ( 2 , 1 , 4 ) , ( 5 , 2 , 4 ) , ( 4 , 5 , 4 ) , ( 0 , 4 , 5 ) , ( 3 , 2 , 5 ) , ( 2 , 5 , 5 ) , ( 5 , 0 , 5 ) } .