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A note on the chromatic number of the square of the Cartesian product of two cycles
Discrete Mathematics 313 (2013) 999–1001
Contents lists available at SciVerse ScienceDirect
Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
Note
A note on the chromatic number of the square of the Cartesian product of two cycles✩ Zehui Shao a,b,∗ , Aleksander Vesel c a
School of Information Science and Technology, Chengdu University, Chengdu, 610106, China
b
Key Laboratory of Pattern Recognition and Intelligent Information Processing, Institutions of Higher Education of Sichuan Province, China
c
Department of Mathematics and Computer Science, PeF, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia
article
info
Article history: Received 7 January 2013 Received in revised form 23 January 2013 Accepted 30 January 2013 Available online 24 February 2013
abstract The square G2 of a graph G is obtained from G by adding edges joining all pairs of nodes at distance 2 in G. In this note we prove that χ ((Cm Cn )2 ) ≤ 6 for m, n ≥ 40. This confirms Conjecture 19 stated in [É. Sopena, J. Wu, Coloring the square of the Cartesian product of two cycles, Discrete Math. 310 (2010) 2327–2333]. © 2013 Elsevier B.V. All rights reserved.
Keywords: Chromatic number Graph labeling Square of a graph Cartesian product
1. Introduction The problem of determining the chromatic number of the square of a graph has attracted lots of attention. In particular, there are many results for planar graphs (see e.g. [2–5]) as well as for the chromatic number of the square of the Cartesian product of paths and cycles [1,4,5]. Let Tm,n = Cm Cn . Sopena and Wu [5] proposed the following conjectures: Conjecture 1. If m, n ≥ 3, then χ (Tm2 ,n ) =
2 )| |V (Tm ,n 2 ) α(Tm ,n
.
Conjecture 2. There exists some constant c such that if m, n ≥ c, then χ (Tm2 ,n ) ≤ 6. In this note, we give a coloring showing that Conjecture 2 holds for c = 40. More formally, we show: Theorem 1. If m, n ≥ 40, then χ (Tm2 ,n ) ≤ 6. In the rest of this section we give necessary definitions, while in the next section Theorem 1 is proved. Let G = (V , E ) be a graph. For two vertices u and v in G let dG (u, v) denote the distance between u and v in G. The square G2 of a graph G is given by V (G2 ) := V (G) and uv ∈ E (G2 ) if and only if 1 ≤ dG (u, v) ≤ 2. The Cartesian product of graphs G and H is the graph GH with vertex set G × H and (x1 , x2 )(y1 , y2 ) ∈ E (GH ) whenever x1 y1 ∈ E (G) and x2 = y2 , or x2 y2 ∈ E (H ) and x1 = y1 . The subgraph of GH induced by u × V (H ) is isomorphic to H. It is called an H-fiber and denoted as H u . ✩ Supported by the Science and Technology Project of Chengdu (RKYB041ZF-023), Sichuan Youth Science & Technology Foundation (2010JQ0032) and the Ministry of Science of Slovenia under the grant 0101-P-297. ∗ Corresponding author at: School of Information Science and Technology, Chengdu University, Chengdu, 610106, China. E-mail addresses: [email protected] (Z. Shao), [email protected] (A. Vesel).
0012-365X/$ – see front matter © 2013 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2013.01.025
1000
Z. Shao, A. Vesel / Discrete Mathematics 313 (2013) 999–1001
A set S of vertices of a graph G is called independent if no two distinct vertices of S are adjacent. The size of the largest independent set in G is called the independence number of G and denoted by α(G). A k-coloring of a graph G is a function f from V (G) onto a set X with k elements such that uv ∈ E (G) implies that f (u) ̸= f (v). The elements of X are called colors. The smallest number k for which a k-coloring exists is the chromatic number χ(G) of G. Finally, for an integer n ≥ 3, a cycle of length n denoted by Cn is a graph whose vertices are 0, 1, . . . , n − 1 and whose edges are pairs i, i + 1, where the arithmetic is done modulo n. 2. Proof We prove Theorem 1 in this section. i+p−1 Let f denote a k-coloring of (Cm Cn )2 . We denote by fi,p the restriction of f to V (Cni ), . . . , V (Cn ). We also write fi for fi,1 . The following lemma plays an important role in the proof. Lemma 1. Let m, n, p ≥ χ((Cm+(t −1)p Cn )2 ) ≤ k.
3, t
≥ 1 and f be a k-coloring of (Cm Cn )2 . If f0,p is a k-coloring of (Cp Cn )2 , then
Proof. Let f ′ be a function from V (Cm+(t −1)p Cn ) onto the set {1, 2, . . . , k} and fi′ the restriction of f ′ to V (Cni ). The function f ′ is defined as follows: fi ′ =
fi , f(i−m) mod p ,
i
In order to see that f ′ is a k-coloring of (Cm+(t −1)p Cn )2 , consider first the vertex (j, m). This vertex is adjacent to (j, m − 2), (j, m − 1), (j − 1, m − 1), (j + 1, m − 1) in the subgraph induced by V (Cn0 ), . . . , V (Cnm−1 ). Note that f ′ (j, m) = f (j, 0). Since (j, 0) is also adjacent to (j, m − 2), (j, m − 1), (j − 1, m − 1), (j + 1, m − 1) in Cn Cm and f is a k-coloring of Cn Cm , this case is settled. The proof for the other vertices of interest: (j, m + 1), (j, m + sp), and (j, m + sp + 1), s ≥ 1, is analogous. Given two integers r and s, let S (r , s) denote the set of all nonnegative integer combinations of r and s: S (r , s) = {α r + β s : α, β ∈ Z + }. We need a result of Sylvester [6]: Lemma 2. If r , s > 1 are relatively prime integers, then t ∈ S (r , s) for all t ≥ (s − 1)(r − 1). We use the following 11 × 11 pattern. 2 4 1 3 6 B= 2 4 1 6 3 5
1 3 6 2 5 1 3 5 4 2 6
4 2 5 1 3 4 2 6 1 5 3
5 1 3 6 2 5 1 4 3 6 2
3 6 2 4 1 3 6 2 5 1 4
2 4 1 3 5 2 4 1 6 3 5
1 3 5 2 6 1 3 5 4 2 6
5 2 6 1 4 5 2 6 3 1 4
6 1 4 5 3 6 1 4 2 5 3
4 5 3 6 2 4 5 3 1 6 2
3 6 2 4 1 3 6 2 5 4 1
2 It is easy to check that B induces a 6-coloring of T11 ,11 . Moreover, the upper five rows of B as well as the first five columns
2 2 2 of B induce a 6-coloring of T11 ,5 (and T5,11 ). Thanks to Lemma 1 we have χ ((C5α+11β C5γ +11δ ) ) ≤ 6 for integers α, β, γ , δ .
Finally, Lemma 2 shows that χ (Tn2,m ) ≤ 6 for every n, m ≥ (11 − 1)(5 − 1) = 40. This assertion completes the proof of the theorem. 3. Conclusion We show that the chromatic number of the square of the Cartesian product Cm Cn is at most 6 if m, n ≥ 40. This result confirms the conjecture of Sopena and Wu [5]. It is known that the chromatic number of (Cm Cn )2 can be larger than 6 for n, m < 40. The graph with the largest known value of min(m, n) that possesses this property is T82,8 (its chromatic number is 7 [5]). Hence, it would be interesting to answer the following: Question 1. What is the smallest c such that if m, n ≥ c, then χ (Tm2 ,n ) ≤ 6?
Z. Shao, A. Vesel / Discrete Mathematics 313 (2013) 999–1001
Acknowledgment We would like to thank Sandi Klavžar for his helpful suggestions. References [1] [2] [3] [4] [5] [6]
S. Chiang, J. Yan, On L(d, 1)-labeling of Cartesian product of a cycle and a path, Discrete Appl. Math. 156 (2008) 2867–2881. J. Heuvel, S. McGuinness, Coloring the square of a planar graph, J. Graph Theory 42 (2003) 110–124. K. Lih, W. Wang, Coloring the square of an outerplanar graph, Taiwanese J. Math. 10 (2006) 1015–1023. E.S. Mahmoodian, F.S. Mousavi, Coloring the square of products of cycles and paths, J. Combin. Math. Combin. Comput. 79 (2011) 101–119. É Sopena, J. Wu, Coloring the square of the Cartesian product of two cycles, Discrete Math. 310 (2010) 2327–2333. J.J. Sylvester, Mathematical questions with their solutions, Educ. Times 41 (1884) 171–178.
1001
Contents lists available at SciVerse ScienceDirect
Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
Note
A note on the chromatic number of the square of the Cartesian product of two cycles✩ Zehui Shao a,b,∗ , Aleksander Vesel c a
School of Information Science and Technology, Chengdu University, Chengdu, 610106, China
b
Key Laboratory of Pattern Recognition and Intelligent Information Processing, Institutions of Higher Education of Sichuan Province, China
c
Department of Mathematics and Computer Science, PeF, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia
article
info
Article history: Received 7 January 2013 Received in revised form 23 January 2013 Accepted 30 January 2013 Available online 24 February 2013
abstract The square G2 of a graph G is obtained from G by adding edges joining all pairs of nodes at distance 2 in G. In this note we prove that χ ((Cm Cn )2 ) ≤ 6 for m, n ≥ 40. This confirms Conjecture 19 stated in [É. Sopena, J. Wu, Coloring the square of the Cartesian product of two cycles, Discrete Math. 310 (2010) 2327–2333]. © 2013 Elsevier B.V. All rights reserved.
Keywords: Chromatic number Graph labeling Square of a graph Cartesian product
1. Introduction The problem of determining the chromatic number of the square of a graph has attracted lots of attention. In particular, there are many results for planar graphs (see e.g. [2–5]) as well as for the chromatic number of the square of the Cartesian product of paths and cycles [1,4,5]. Let Tm,n = Cm Cn . Sopena and Wu [5] proposed the following conjectures: Conjecture 1. If m, n ≥ 3, then χ (Tm2 ,n ) =
2 )| |V (Tm ,n 2 ) α(Tm ,n
.
Conjecture 2. There exists some constant c such that if m, n ≥ c, then χ (Tm2 ,n ) ≤ 6. In this note, we give a coloring showing that Conjecture 2 holds for c = 40. More formally, we show: Theorem 1. If m, n ≥ 40, then χ (Tm2 ,n ) ≤ 6. In the rest of this section we give necessary definitions, while in the next section Theorem 1 is proved. Let G = (V , E ) be a graph. For two vertices u and v in G let dG (u, v) denote the distance between u and v in G. The square G2 of a graph G is given by V (G2 ) := V (G) and uv ∈ E (G2 ) if and only if 1 ≤ dG (u, v) ≤ 2. The Cartesian product of graphs G and H is the graph GH with vertex set G × H and (x1 , x2 )(y1 , y2 ) ∈ E (GH ) whenever x1 y1 ∈ E (G) and x2 = y2 , or x2 y2 ∈ E (H ) and x1 = y1 . The subgraph of GH induced by u × V (H ) is isomorphic to H. It is called an H-fiber and denoted as H u . ✩ Supported by the Science and Technology Project of Chengdu (RKYB041ZF-023), Sichuan Youth Science & Technology Foundation (2010JQ0032) and the Ministry of Science of Slovenia under the grant 0101-P-297. ∗ Corresponding author at: School of Information Science and Technology, Chengdu University, Chengdu, 610106, China. E-mail addresses: [email protected] (Z. Shao), [email protected] (A. Vesel).
0012-365X/$ – see front matter © 2013 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2013.01.025
1000
Z. Shao, A. Vesel / Discrete Mathematics 313 (2013) 999–1001
A set S of vertices of a graph G is called independent if no two distinct vertices of S are adjacent. The size of the largest independent set in G is called the independence number of G and denoted by α(G). A k-coloring of a graph G is a function f from V (G) onto a set X with k elements such that uv ∈ E (G) implies that f (u) ̸= f (v). The elements of X are called colors. The smallest number k for which a k-coloring exists is the chromatic number χ(G) of G. Finally, for an integer n ≥ 3, a cycle of length n denoted by Cn is a graph whose vertices are 0, 1, . . . , n − 1 and whose edges are pairs i, i + 1, where the arithmetic is done modulo n. 2. Proof We prove Theorem 1 in this section. i+p−1 Let f denote a k-coloring of (Cm Cn )2 . We denote by fi,p the restriction of f to V (Cni ), . . . , V (Cn ). We also write fi for fi,1 . The following lemma plays an important role in the proof. Lemma 1. Let m, n, p ≥ χ((Cm+(t −1)p Cn )2 ) ≤ k.
3, t
≥ 1 and f be a k-coloring of (Cm Cn )2 . If f0,p is a k-coloring of (Cp Cn )2 , then
Proof. Let f ′ be a function from V (Cm+(t −1)p Cn ) onto the set {1, 2, . . . , k} and fi′ the restriction of f ′ to V (Cni ). The function f ′ is defined as follows: fi ′ =
fi , f(i−m) mod p ,
i
In order to see that f ′ is a k-coloring of (Cm+(t −1)p Cn )2 , consider first the vertex (j, m). This vertex is adjacent to (j, m − 2), (j, m − 1), (j − 1, m − 1), (j + 1, m − 1) in the subgraph induced by V (Cn0 ), . . . , V (Cnm−1 ). Note that f ′ (j, m) = f (j, 0). Since (j, 0) is also adjacent to (j, m − 2), (j, m − 1), (j − 1, m − 1), (j + 1, m − 1) in Cn Cm and f is a k-coloring of Cn Cm , this case is settled. The proof for the other vertices of interest: (j, m + 1), (j, m + sp), and (j, m + sp + 1), s ≥ 1, is analogous. Given two integers r and s, let S (r , s) denote the set of all nonnegative integer combinations of r and s: S (r , s) = {α r + β s : α, β ∈ Z + }. We need a result of Sylvester [6]: Lemma 2. If r , s > 1 are relatively prime integers, then t ∈ S (r , s) for all t ≥ (s − 1)(r − 1). We use the following 11 × 11 pattern. 2 4 1 3 6 B= 2 4 1 6 3 5
1 3 6 2 5 1 3 5 4 2 6
4 2 5 1 3 4 2 6 1 5 3
5 1 3 6 2 5 1 4 3 6 2
3 6 2 4 1 3 6 2 5 1 4
2 4 1 3 5 2 4 1 6 3 5
1 3 5 2 6 1 3 5 4 2 6
5 2 6 1 4 5 2 6 3 1 4
6 1 4 5 3 6 1 4 2 5 3
4 5 3 6 2 4 5 3 1 6 2
3 6 2 4 1 3 6 2 5 4 1
2 It is easy to check that B induces a 6-coloring of T11 ,11 . Moreover, the upper five rows of B as well as the first five columns
2 2 2 of B induce a 6-coloring of T11 ,5 (and T5,11 ). Thanks to Lemma 1 we have χ ((C5α+11β C5γ +11δ ) ) ≤ 6 for integers α, β, γ , δ .
Finally, Lemma 2 shows that χ (Tn2,m ) ≤ 6 for every n, m ≥ (11 − 1)(5 − 1) = 40. This assertion completes the proof of the theorem. 3. Conclusion We show that the chromatic number of the square of the Cartesian product Cm Cn is at most 6 if m, n ≥ 40. This result confirms the conjecture of Sopena and Wu [5]. It is known that the chromatic number of (Cm Cn )2 can be larger than 6 for n, m < 40. The graph with the largest known value of min(m, n) that possesses this property is T82,8 (its chromatic number is 7 [5]). Hence, it would be interesting to answer the following: Question 1. What is the smallest c such that if m, n ≥ c, then χ (Tm2 ,n ) ≤ 6?
Z. Shao, A. Vesel / Discrete Mathematics 313 (2013) 999–1001
Acknowledgment We would like to thank Sandi Klavžar for his helpful suggestions. References [1] [2] [3] [4] [5] [6]
S. Chiang, J. Yan, On L(d, 1)-labeling of Cartesian product of a cycle and a path, Discrete Appl. Math. 156 (2008) 2867–2881. J. Heuvel, S. McGuinness, Coloring the square of a planar graph, J. Graph Theory 42 (2003) 110–124. K. Lih, W. Wang, Coloring the square of an outerplanar graph, Taiwanese J. Math. 10 (2006) 1015–1023. E.S. Mahmoodian, F.S. Mousavi, Coloring the square of products of cycles and paths, J. Combin. Math. Combin. Comput. 79 (2011) 101–119. É Sopena, J. Wu, Coloring the square of the Cartesian product of two cycles, Discrete Math. 310 (2010) 2327–2333. J.J. Sylvester, Mathematical questions with their solutions, Educ. Times 41 (1884) 171–178.
1001