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 Zm.
• 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 χk(G).

• Let Gk be the kth graph in which its vertex set is V(G) and

there is an edge between two vertices of Gk if and only if they have distance at most k.

• Hence χk(G) is equal to χ(Gk).

Cartesian product of graphs

• Let G1 and G2 be graphs.
• We consider the usual Cartesian (or box) product G1G2.
• The vertex set of G1G2 is the Cartesian product

V(G1) × V(G2) of V(G1) and V(G2).

• There is an edge between two vertices of G1G2 if and
• nly if they are adjacent in exactly one coordinate and

agree in the other.

2-distance coloring of hypercube Qn

• The graph for the k-distance coloring related to binary

codes is Hamming graph H(2, n) (or hypercube Qn = P2P2 · · · P2).

2-distance coloring of hypercube Qn

• The graph for the k-distance coloring related to binary

codes is Hamming graph H(2, n) (or hypercube Qn = P2P2 · · · P2).

• The 2-distance coloring of Qn has been studied by

Fertin-Godard-Raspaud (’03), Jamison-Matthews-Villalpando (’06), Kim-Du-Pardolos (’00), Ngo-Du-Graham (’00), ¨ Osterg˚ ard (’04).

• Although Qn is a very simple graph, the exact 2-distance

chromatic number of Qn is known only for some n.

• P

.R.J. ¨ Osterg˚ ard [On a hypercube coloring problem, J.

• Combin. Theory Ser. A, 108 (2004) 199–204].

limn→∞

χ2(Qn) n

= 1 and limn→∞

χ3(Qn) n

= 2.

2-distance coloring of Cm1Cm2

• The graph for the k-distance coloring related to non-binary

codes is the Cartesian product cycles.

2-distance coloring of Cm1Cm2

• 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 χ2(Cm1Cm2),

where Cm1 and Cm2 are cycles of lengths m1 and m2, respectively.

• They proved χ2(Cm1Cm2) ≤ 7 except when

(m1, m2) = (3, 3) in which case the value is 9, and when (m1, m2) = (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

• f d cycles is at most 6d + O(log d).

Connection with codes over Zn

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 Zn

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 χ2(CmCmCm) for a positive integer m ≥ 3.

Representation of G(n, m) as Zn

m

• Let G(n, m) := CmCm · · · Cm.
• We may represent G(n, m) as Zn

m, where Zm is the ring of

integers modulo m.

• The Lee weight wtL(u) of u = (u1, . . . , un) ∈ Zn

m is defined

as wtL(u) = n

i=1 min{ui, m − ui}.

• The Lee distance dL(u, v) of u = (u1, . . . , un) and

u = (v1, . . . , vn) is wtL(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 Zn

m.

• This shows that there is an edge between u and v in

G(n, m) if and only if their Lee distance dL(u, v) is 1.

• This correspondence connects coloring problems with

coding theory problems with respect to the Lee weight.

Lower bounds for χ2(G(n, m))

As the degree of any vertex of G(n, m) is 2n, we have a trivial lower bound below. Lemma 1 χ2(G(n, m)) ≥ 2n + 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 χ2(G(n, m)) = 2n + 1.

Corollary 1 If 2n + 1 divides m, then χ2(G(n, m)) = 2n + 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 Zn

m if 2n + 1|m and conjectured that there is no perfect

t-error-correcting Lee code in Zn

m if n > 2 and t > 1. Thus the

claim follows from Theorem 1. Corollary 2 If χ2(G(n, m)) = 2n + 1, then 2n + 1 divides mn. Proof If χ2(G(n, m)) = 2n + 1, then there exists a perfect code C on G(m, n) by Theorem 1. Thus (2n + 1) · |C| = mn, that is, 2n + 1 divides mn, as required.

Sphere packing bound for codes in Zn

m

Lemma (the sphere packing bound) Let AL

m(n, d) be the size of an optimal code in Zn m with Lee

distance d. Then AL

m(n, d) ≤

• mn

Vm(n, e)

• ,

where e = ⌊(d − 1)/2⌋ and Vm(n, e) is the volume of the ball of radius e around any vertex of Zn

m.

In particular, if n = 3 and d = 3, then AL

m(3, 3) ≤ ⌊m3/7⌋.

Furthermore, if G = CmCmCm, then α(G2) = AL

m(3, 3) and

⌈m3/α(G2)⌉ ≤ χ(G2) = χ2(G), where α(G2) is the size of maximum independent set in G2.

Theorem 2 χ2(C3C3C3) = 9 Proof Let G = C3C3C3. We show that α(G2) = 3. By the sphere packing bound, one has AL

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 AL

3(3, 3) ≥ 3. Hence α(G2) = AL 3(3, 3) = 3.

Therefore χ2(C3C3C3) ≥ 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 χ2(C3C3C3) ≤ 9. Therefore we have χ2(C3C3C3) = 9.

Lemma 2 AL

4(3, 3) = 8

Proof Let C4 = {(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(C4) = 3. Note that C4 is invariant under the cyclic shift. Hence AL

4(3, 3) ≥ 8.

Now we will show that AL

4(3, 3) ≤ 8. Let C ⊂ Z3 4 be a 4-ary code

with d(C) = 3. Put Rj := {y = (y1, y2, y3) ∈ Z3

4 | y1 = j} and

Cj := C ∩ Rj for j = 0, . . . , 3. Then for each j, |Cj| ≤ 3 since AL

4(2, 3) ≤ ⌊ 16 5 ⌋ = 3 by the sphere packing bound. However we

show that |Cj| = 3 for any j. Suppose that |Cj| = 3 for some j. We may assume that (j, 0, 0) ∈ Cj. Then we can check that Cj ⊂ {(j, 0, 0), (j, 1, 2), (j, 2, 1), (j, 2, 2), (j, 2, 3), (j, 3, 2)}. Since d(Cj) = 3, we have |Cj| ≤ 2. Therefore |C| ≤ 3

j=0 |Cj| ≤ 4 · 2 = 8, completing the proof.

Theorem 3 χ2(C4C4C4) = 8 Proof Let G = C4C4C4. Then α(G2) = 8 by Lemma 2; hence χ2(G) ≥ 8. To show that the bound is tight, we use the code C4 in Lemma 2. Just as in Theorem 2, each ball of radius

• ne around a codeword of C4 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 χ2(G) = 8.

Theorem 4 χ2(C5C5C5) = 9 Proof Let G = C5C5C5. It is known that AL

5(3, 3) = 15 (see

Quistorff, 2006). Hence α(G2) = 15; hence χ2(G) ≥ 9. Showing that χ2(G) ≤ 9 is technical, hence omitted.

AL

6(3, 3) = 26

What is the size of an optimal code in Z3

6 with d = 3, that is,

AL

6(3, 3)?

AL

6(3, 3) = 26

What is the size of an optimal code in Z3

6 with d = 3, that is,

AL

6(3, 3)?

By the Sphere-Packing bound, AL

6(3, 3) ≤ 30 (see also

Quistorff, 2006). For the first time, we show that AL

6(3, 3) = 26.

Lemma 3 AL

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)}.

Theorem 5 χ2(C6C6C6) = 9 Proof Let G = C6C6C6. We have AL

6(3, 3) = 26 by Lemma

• 3. Hence χ2(G) ≥

|V(G)| |AL

6(3,3)| = 216

26 > 8. Hence χ2(G) ≥ 9.

On the other hand, using χ2(C3C3C3) = 9 (Theorem 2), we have χ2(G) ≤ 9. Hence χ2(G) = 9.

Theorem 6 χ2(CmCmCm) = 7 if and only if m is a multiple of 7. Proof Suppose that χ2(CmCmCm) = 7. Then by Theorem 1, there exists a perfect code C in G(3, m). Thus 7 · |C| = m3. Hence 7 divides m. Conversely suppose that 7|m and let m = 7m′. If m = 7 then there is a perfect code C in G(3, 7) by Golomb and Welch (1970). We copy C in the x, y, and z directions in G(3, m) with a total of (m′)3 times. Then we obtain a perfect code C′ in G(3, m). Thus χ2(CmCmCm) = 7 by Theorem 1.

Corollary 2 For any positive k, χ2(CmCmCm) = 8 if m = 4k, and χ2(CmCmCm) ≤ 9 if m = 3k or m = 5k. Proof In this proof, let Gm := G(3, m) = CmCmCm. First, for each m = 3k, we converts numbers in {0, 1, 2, . . . , m − 1} by g(x) ≡ x (mod 3) where g : {0, 1, 2, . . . , m − 1} → {0, 1, 2}. Let h : V(C3C3C3) → {0, 1, . . . , 8} be a proper 9-coloring of G2

3.

Then for each vertex (x1, x2, x3) ∈ V(G3k), assign f((x1, x2, x3)) = h((g(x1), g(x2), g(x3)). Then f is a proper 9-coloring of G2

• 3k. Hence χ2(G3k) ≤ 9. Similarly we can show

that χ2(G5k) ≤ 9 and χ2(G4k) = 8.

Conjecture

• E. Sopena and J. Wu [Coloring the square of the Cartesian

product of two cycles, Discrete Math.] conjectured χ2(G) = ⌈ |V(G)|

α(G2) ⌉ for G = CmCn.

We also conjecture the following. Conjecture Let m ≥ 3 be a positive integer and Gm = CmCmCm. Then χ2(Gm) = |V(Gm)| α(G2

m)

• where α(G2

m) is the size of maximum independent set in G2 m.

In this talk, we have shown that this conjecture is true for 3 ≤ m ≤ 7.

Conclusion

• In this talk, we have introduced an interesting connection

between k-distance coloring and codes over Zm.

• We have computed exact values of χ2(G(3, m)) for

3 ≤ m ≤ 8 and m = 4k where k is a positive integer, and an upper bound if m = 3k or 5k.

• In general, it seems interesting and difficult (?) to study

k-distance coloring on the Johnson graph with its connection to constant weight codes. THANK YOU FOR YOUR ATTENTION!