On the locating chromatic number of Kneser graphs

Discrete Applied Mathematics 159 (2011) 2214–2221 Contents lists available at SciVerse ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam On the locating chromatic number of Kneser graphs Ali Behtoei, Behnaz Omoomi ∗ Department of Mathematical Sciences, Isfahan University of Technology, 84156-83111, Isfahan, Iran article info Article history: Received 8 August 2010 Received in revised form 8 June 2011 Accepted 13 July 2011 Available online 20 August 2011 Keywords: Kneser graph Locating coloring Locating chromatic number Metric dimension abstract Let c be a proper k-coloring of a connected graph G and Π = (C1 , C2 , . . . , Ck ) be an ordered partition of V (G) into the resulting color classes. For a vertex v of G, the color code of v with respect to Π is defined to be the ordered k-tuple cΠ (v) := (d(v, C1 ), d(v, C2 ), . . . , d(v, Ck )), where d(v, Ci ) = min{d(v, x)|x ∈ Ci }, 1 ≤ i ≤ k. If distinct vertices have distinct color codes, then c is called a locating coloring. The minimum number of colors needed in a locating coloring of G is the locating chromatic number of G, denoted by χL (G). In this paper, we study the locating chromatic number of Kneser graphs. First, among some other results, we show that χL (KG(n, 2)) = n − 1 for all n ≥ 5. Then, we prove that χL (KG(n, k)) ≤ n − 1, when n ≥ k2 . Moreover, we present some bounds for the locating chromatic number of odd graphs. © 2011 Elsevier B.V. All rights reserved. 1. Introduction Let G be a graph without loops and multiple edges with vertex set V (G) and edge set E (G). A proper k-coloring of G is a function c defined from V (G) onto a set of colors C = {1, 2, . . . , k} such that every two adjacent vertices have different colors. In fact, for every i, 1 ≤ i ≤ k, the set c −1 (i) is a nonempty independent set of vertices which is called the color class i. The minimum cardinality k for which G has a proper k-coloring is the chromatic number of G, denoted by χ(G). For a connected graph G, the distance d(u, v) between two vertices u and v in G is the length of a shortest path between them, and for a subset S of V (G), the distance between u and S is given by d(u, S ) := min{d(u, x) | x ∈ S }. A set W ⊆ V (G) is called a resolving set, if for each two distinct vertices u, v ∈ V (G) there exists w ∈ W such that d(u, w) ̸= d(v, w), see [9,16]. A resolving set with the minimum cardinality is called a metric basis and its cardinality is called the metric dimension of G, denoted by dimM (G). Definition 1 ([2]). Let c be a proper k-coloring of a connected graph G and Π = (C1 , C2 , . . . , Ck ) be an ordered partition of V (G) into the resulting color classes. For a vertex v of G, the color code of v with respect to Π is defined to be the ordered k-tuple cΠ (v) := (d(v, C1 ), d(v, C2 ), . . . , d(v, Ck )). If distinct vertices of G have distinct color codes, then c is called a locating coloring of G. The locating chromatic number, χL (G), is the minimum number of colors in a locating coloring of G. The concept of locating coloring was first introduced by Chartrand et al. in [2] and studied further in [1,3]. This concept has been called with the other names such as resolving coloring and independent resolving partition, see [13]. Note that, since every locating coloring is a proper coloring, χ(G) ≤ χL (G). For more results on the subject and related subjects, one can see [1–6,13,14]. ∗ Corresponding author. E-mail addresses: alibehtoei@math.iut.ac.ir (A. Behtoei), bomoomi@cc.iut.ac.ir (B. Omoomi). 0166-218X/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.dam.2011.07.015 2215 A. Behtoei, B. Omoomi / Discrete Applied Mathematics 159 (2011) 2214–2221 We have the following two theorems for the relation between the locating chromatic number of a graph with its diameter, metric dimension and chromatic number. Theorem A ([2]). If G is a connected graph with diameter d and locating chromatic number l, then |V (G)| ≤ ldl−1 . Theorem B ([2]). For each connected graph G with at least three vertices, χL (G) ≤ χ(G) + dimM (G). Hereafter, we denote the set {1, 2, . . . , n} by [n] and the collection of all k-subsets of the set [n] by  [n] k  . Let n and k be twopositive integers. The Kneser graph with parameters n and k, n ≥ 2k, denoted by KG(n, k), is the graph with vertex  set [n] k such that two vertices are adjacent if and only if the corresponding subsets are disjoint. Let k ≥ 3. Kneser graph KG(2k, k) is a matching and the smallest positive integer n for which KG(n, k) is connected, is n = 2k + 1. Kneser graphs KG(2k + 1, k), k ≥ 3, are known as the odd graphs. The distance between two vertices in Kneser graph and the diameter of this graph are investigated in [18]. We summarize these results in the following theorem. Theorem C ([18]). Let A, B ∈  [n] k  be two different vertices of Kneser graph KG(n, k), where n ≥ 2k + 1. If |A ∩ B| = s, then the distance d(A, B) in KG(n, k) is given by   d(A, B) = min 2 k−s n − 2k  ,2 Moreover, the diameter of KG(n, k) is   s n − 2k k−1 n−2k    +1 . + 1. Kneser graphs have many interesting properties and have been the subject of many researches. It was conjectured by Kneser in 1955 [11] and proved by Lovász in 1978 [12] that χ(KG(n, k)) = n − 2k + 2. Since then, several types of colorings of Kneser graphs have been considered. For example, the circular chromatic number, the b-chromatic number and the multichromatic number of Kneser graphs were investigated in [8,10,17], respectively. In this paper, we study the locating chromatic number of Kneser graphs. In the next section, among some other results, we show that χL (KG(n, 2)) = n − 1 for n ≥ 5. Then, we prove that χL (KG(n, k)) ≤ n − 1 when n ≥ k2 . For the case k = 3, we show that this inequality holds for every positive integer n ≥ 7. In the last section, we provide a lower bound for the locating chromatic number, an upper bound for the metric dimension and accordingly for the locating chromatic number of odd graphs. Through the paper, for convenience, we denote the vertex {i1 , i2 , . . . , ik } in KG(n, k) by i1 i2 . . . ik . 2. The locating chromatic number of KG (n, 2) If A is an independent set in KG(n, 2), then either all vertices in A have a common element of [n], say a, or A = {ab, ac , bc } for some a, b, c ∈ [n]. Since each vertex ij in KG(n, 2) corresponds to the edge ij in Kn , an independent set in KG(n, 2) corresponds to a star subgraph or a triangle subgraph in the complete graph Kn . From now on, we call an independent set in KG(n, 2) of the first form starlike with center a, and of the second form triangular. Since every proper coloring is a partition of vertices into independent sets, it is easy to see that every proper coloring of the Kneser graph KG(n, 2) is equivalent to an edge decomposition of the complete graph Kn into star and triangle subgraphs. In order to study the locating chromatic number of KG(n, 2), we need the following theorem. A biclique partition of a graph G is a partition of the edge set of G into complete bipartite graphs. Since a single edge can form a biclique, every graph has a biclique partition. The biclique partition number bp(G) of G is the smallest number of bicliques that partition G. Since the complete graph Kn can be partitioned into n − 1 stars, bp(Kn ) ≤ n − 1. In fact, we have the following famous theorem. Theorem D ([7]). The biclique partition number of the complete graph Kn is n − 1. Consider the Kneser graph KG(n, k), n > 2k. Let n = 2k + d, d ≥ 1. There is a proper coloring of KG(n, k) with χ(KG(n, k)) = d + 2 colors as follows. For i = 1, 2, . . . , d + 1, let Ci consist of all k-subsets of [n] which contain i as the smallest element. The remaining k-subsets are contained in the set {d + 2, d + 3, . . . , d + 2k}, which has only 2k − 1 elements. Hence, they all intersect (are non adjacent). Thus, we can use color d + 2 for all of them. For the case k = 2, χ(KG(n, 2)) = n − 2 and the latter color class in the above proper coloring is of triangular form. In this section, we first show that all proper (n − 2)-colorings of KG(n, 2) are similar to the proper coloring given above. Next, we determine the exact value of the locating chromatic number of KG(n, 2). Theorem 1. In every proper (n − 2)-coloring of the Kneser graph KG(n, 2), n ≥ 5, there exists a unique triangular color class. Furthermore, if c is a proper (n − 2)-coloring of KG(n, 2), then by renaming the symbols 1, 2, . . . , n, if it is necessary, we have the color classes F1 , F2 , . . . , Fn−2 with the following properties. (a) Fn−2 = {n(n − 1), n(n − 2), (n − 1)(n − 2)}, i.e. Fn−2 is triangular; (b) for each i, 1 ≤ i ≤ n − 3 , Fi is starlike with center i, and {in, i(n − 1), i(n − 2)} ⊆ Fi ; (c) each vertex ij with {i, j} {n, n − 1, n − 2} = ∅, is either in Fi or Fj . 2216 A. Behtoei, B. Omoomi / Discrete Applied Mathematics 159 (2011) 2214–2221 Proof. We will prove the theorem by induction on n. Let n = 5 and consider a proper 3-coloring of KG(n, 2) with color classes F1 , F2 and F3 . Equivalently, we have an edge decomposition of the complete graph K5 into stars and triangles. By Theorem D, there exists no edge decomposition of K5 into three star subgraphs. Thus, at least one of the color classes is triangular, say F3 = {34, 35, 45}. Now, six vertices 13, 14, 15; 23, 24, 25 should be distributed between two color classes F1 and F2 . Since F1 and F2 are independent sets, the only possibility is, say {13, 14, 15} ⊆ F1 and {23, 24, 25} ⊆ F2 , which means F1 and F2 are starlike with centers 1 and 2, respectively. The remaining vertex 12 can be either in F1 or F2 . Hence, the theorem holds for the induction basis. Now, let n ≥ 6 and suppose that c is a proper (n − 2)-coloring of KG(n, 2). Equivalently, we have an edge decomposition of the complete graph Kn into stars and triangles. Similar to the one above, by Theorem D, we have at least one triangular color class, say Fn−2 = {n(n − 1), n(n − 2), (n − 1)(n − 2)}. Let Xn := n−3  i=1 {in}, Xn−1 := n−3  i=1 {i(n − 1)}, Xn−2 := n−3  i=1 {i(n − 2)} and X := Xn ∪ Xn−1 ∪ Xn−2 . Note that, each Xi is starlike, |X | = 3(n − 3), and for each i, j, n − 2 ≤ i < j ≤ n, the induced subgraph of KG(n, 2) on Xi ∪ Xj is a complete bipartite graph with a perfect matching deleted. The vertices in X should be distributed in n − 3 color classes F1 , F2 , . . . , Fn−3 . Thus, there exists a color class, say F1 , which contains at least three vertices of X . Claim. There exists no k, n − 2 ≤ k ≤ n, such that F1 ∩ X ⊆ Xk . Proof of claim. Assume to the contrary, and without loss of generality, that F1 ∩ X ⊆ Xn . Since F1 is an independent set and |F1 ∩ X | ≥ 3, F1 is starlike with center n. Now, the 2(n − 3) = 2n − 6 vertices of Xn−1 ∪ Xn−2 should be distributed in the n − 4 color classes F2 , F3 , . . . , Fn−3 . Hence, there exists a color class, say F2 , which contains at least three vertices of Xn−1 ∪ Xn−2 . Since each vertex i(n − 1) in Xn−1 is adjacent to all of the vertices in Xn−2 except the vertex i(n − 2), the only possibility is F2 ∩ (Xn−1 ∪ Xn−2 ) ⊆ Xn−1 or F2 ∩ (Xn−1 ∪ Xn−2 ) ⊆ Xn−2 . Without loss of generality, we can assume that F2 ∩ (Xn−1 ∪ Xn−2 ) ⊆ Xn−1 . Similarly, this implies that F2 is starlike with center n − 1. Now, for each i, 3 ≤ i ≤ n − 3, let F̄i := Fi and also let F̄1 := F1 ∪ {(n − 1)n, (n − 2)n}, F̄2 := F2 ∪ {(n − 1)(n − 2)}. Thus, each F̄i is an independent set in the Kneser graph KG(n, 2). This means that we have a proper (n − 3)-coloring of KG(n, 2), which is a contradiction. Thus, the claim is proved. If |F1 ∩ Xk | ≥ 3 for some k, n − 2 ≤ k ≤ n, then F1 is starlike with center k. Thus, F1 ∩ X ⊆ Xk , which is impossible by the above claim. If for some k, n − 2 ≤ k ≤ n, |F1 ∩ Xk | = 2, then there exist i and j, 1 ≤ i < j ≤ n − 3, such that F1 ∩ Xk = {ik, jk}. Since F1 contains at least three vertices from X , F1 is an independent set and every vertex in X \ Xk is adjacent to at least one of the vertices ik or jk, we should have F1 ∩ X ⊆ Xk , which by claim is impossible. Therefore, |F1 ∩ Xk | = 1 for each k, n − 2 ≤ k ≤ n. By renaming the symbols 1, 2, . . . , n − 3, if it is necessary, assume that F1 ∩ Xn = {1n}. Since F1 is an independent set and the vertex 1n is adjacent to all of the vertices in Xn−1 ∪ Xn−2 except 1(n − 1) and 1(n − 2), we have F1 ∩ X = {1n, 1(n − 1), 1(n − 2)}. This implies that F1 is starlike with center 1. Let F̄1 := {12, 13, . . . , 1n} and for each 2 ≤ j ≤ n − 2, let F̄j := Fj \ F̄1 . Note that, F̄1 is starlike with center 1 and |F̄1 | = n − 1, F̄j ⊆ Fj for each 2 ≤ j ≤ n − 3, and F̄n−2 = Fn−2 . Since F̄1 , F̄2 , . . . , F̄n−2 is a partition of the vertices of KG(n, 2) into independent sets and χ(KG(n, 2)) = n − 2, none of the F̄j ’s is an empty set.  Now,  we can consider F̄2 , F̄3 , . . . , F̄n−2 as a proper (n − 3)- coloring of the Kneser graph KG(n − 1, 2) with vertex set [n]\{1} 2 . By the induction hypothesis, exactly one of these color classes is triangular, which is F̄n−2 , and by renaming F̄j ’s, if it is necessary, each F̄j is a starlike independent set with center j, 2 ≤ j ≤ n − 3. Moreover, by the induction hypothesis, we have {jn, j(n − 1), j(n − 2)} ⊆ F̄j ⊆ Fj , 2 ≤ j ≤ n − 3. Thus, each Fj is a starlike independent set with center j. Using the induction hypothesis, it is easy to see that the condition (c) also holds.  By Theorem 1, every optimal proper coloring of KG(n, 2) has a unique triangular color class. This set uses exactly three n symbols from the set [n] which can be chosen in 3 different ways. If we choose this triangular set, then the colors of n 3(n − 3) vertices will be determined. Now, the remaining 2 − 3 − 3(n − 3) vertices should be distributed among n − 3 starlike color classes and each of them has two choices for their color. Accordingly, we have the following corollary. 2217 A. Behtoei, B. Omoomi / Discrete Applied Mathematics 159 (2011) 2214–2221 Corollary 1. The number of different optimal proper colorings of the Kneser graph KG(n, 2) is n 3 n 2( 2 )−3(n−2) . Now, we are ready to determine the exact value of the locating chromatic number of KG(n, 2). Theorem 2. For all positive integers n ≥ 5, we have χL (KG(n, 2)) = n − 1. Proof. By Theorem C, the diameter of KG(n, 2) is two. First, we show that χL (KG(n, 2)) ≥ n − 1. Since n − 2 = χ(KG(n, 2)) ≤ χL (KG(n, 2)), it is sufficient to show that no proper (n − 2)-coloring of KG(n, 2) is a locating coloring. Let c be a proper (n − 2)coloring of KG(n ∏, 2). By the same notations and assumptions as in Theorem 1, assume that F1 , F2 , . . . , Fn−2 are its color classes and let = (F1 , . . . , Fn−2 ) be an ordered partition of the vertex set into the resulting color classes. Now, since d(1n, (n − 1)(n − 2)) = d(1(n − 1), (n − 2)n) = 1, and for each j, 2 ≤ j ≤ n − 3, d(1n, j(n − 1)) = d(1(n − 1), j(n − 2)) = 1, we have cΠ (1n) = cΠ (1(n − 1)) = (0, 1, 1, . . . , 1). Thus, c is not a locating coloring. Now, to complete the proof, we provide a locating (n − 1)-coloring of KG(n, 2). Let C1 := {1j | 2 ≤ j ≤ n − 2} ∪ {1n}, Cn−1 := {1(n − 1), (n − 1)n}, and for each i, 2 ≤ i ≤ n − 2, let Ci := {ij | i < j ≤ n}. Note that, each Ci , 1 ≤ i ≤ n − 1, is starlike with center i and C1 , C2 , . . . , Cn−1 is a partition of the vertex set. Therefore, by assigning ∏ color i to the vertices in Ci , we get a proper (n − 1)coloring of KG(n, 2). Let us show that this is a locating coloring. Let = (C1 , . . . , Cn−1 ) be an ordered partition of the vertex set into the resulting color classes. It is sufficient to show that distinct vertices of each color class have distinct color codes. Since d(1(n − 1), C1 ) = 2 and d((n − 1)n, C1 ) = 1, this holds for the color class Cn−1 . This also holds for the color class Cn−2 , since d((n − 2)(n − 1), Cn−1 ) = 2, d((n − 2)n, Cn−1 ) = 1. Now, consider color class Ci , 1 ≤ i ≤ n − 3, and let ij and ik be two distinct vertices in Ci with j < k. We know that d(ij, Cj ) = 2. If k = n and j < n − 1, then d(ik, j(j + 1)) = 1, where j(j + 1) ∈ Cj . If k = n and j = n − 1, then i ̸= 1, and d(ik, 1(n − 1)) = 1, where 1(n − 1) ∈ Cj . If k ̸= n, then d(ik, jn) = 1, where jn ∈ Cj . Therefore, in all cases we have d(ik, Cj ) = 1, which implies cΠ (ij) ̸= cΠ (ik).  3. The locating chromatic number of KG (n, k ) In this section, we give an upper bound for the locating chromatic number of some class of Kneser graphs. Note that, an independent set in KG(n, k) is called starlike with center j, whenever j is the unique common symbol of every two vertices in it. Theorem 3. Let n, k be two positive integers, where k ≥ 3. If n ≥ k2 , then χL (KG(n, k)) ≤ n − 1. Proof. Since n ≥ k2 and k ≥ 3, by Theorem C, the diameter of KG(n, k) is two. To prove the theorem, it is enough to provide a locating  (n − 1)-coloring for KG(n, k). For this purpose, we first construct a family {F2 , F3 , . . . , Fn } of the subsets of the vertex set [n] k with the following properties. (a) Each Fj is starlike with center j which contains exactly k + 1 vertices, and for each two distinct vertices X , Y ∈ Fj , X ∩ Y = {j}. (b) Each Fj uses exactly 1 + (k − 1)(k + 1) = k2 symbols of the set [n], i.e. there exists a k2 -element subset B of [n] such that each vertex in Fj is a subset of B. (c) For each 2 ≤ r < s ≤ n, Fr ∩ Fs =  ∅.  (d) If 2 ≤ j ≤ n, then each vertex A ∈ [n] k \ Fj with j ̸∈ A, is adjacent to at least one vertex in Fj and d(A, Fj ) = 1. If k = 3, then we define the family {F2 , F3 , . . . , Fn } as follows. F2 := {259, 268, 247, 213}, F3 := {324, 319, 358, 367}, F4 := {413, 456, 479, 428}, F5 := {512, 534, 567, 589}, F6 := {612, 634, 658, 679}, F7 := {712, 734, 759, 768}, F8 := {812, 834, 857, 869}, and for each l, 9 ≤ l ≤ n, let Fl := {l12, l34, l56, l78}. It is easy to check that the properties (a) to (d) hold for this family. If k ≥ 4, then we construct the family {F2 , F3 , . . . , Fn } as follows. Let A2,1 , A2,2 , . . . , A2,k+1 be a partition of the set [k2 ] \ {2} into k + 1 subsets, each of size k − 1. Note that this can be (k2 −1)! done in ((k−1)!)k+1 (k+1)! different ways which is strictly greater than one, since k > 2. Now, let F2 := k+1  j=1 {A2,j ∪ {2}}. 2218 A. Behtoei, B. Omoomi / Discrete Applied Mathematics 159 (2011) 2214–2221 In fact, F2 is an independent subset of the vertex set  [n] k  of size k + 1 which is starlike with center 2. Each vertex in F2 is a subset of [k2 ] and X ∩ Y = {2} for each two distinct vertices X , Y ∈ F2 . If A ∈  [n] k  \ F2 is a vertex with 2 ̸∈ A, then it is adjacent to at least one vertex in F2 , thusd(A, F2 ) = 1. In an inductive way, suppose that F2 , F3 , . . . , Fi , i < k2 , are constructed in such a way that each Fj ⊆  [k2 ] k , 2 ≤ j ≤ i, is starlike of size k + 1 with center j, X ∩ Y = {j} for each two distinct vertices X , Y ∈ Fj , and Fr Fs = ∅ whenever r ̸= s. Then, we construct Fi+1 as follows. For each j, 2 ≤ j ≤ i, let Vj be the unique vertex in Fj which contains the element i + 1, and let Uj := Vj \ {i + 1}. The number of different partitions of the set [k2 ] \ {i + 1} into k + 1 subsets of size k − 1 in which Uj appears as a partition part of them, is equal to the number of different partitions of the set [k2 ] \ (Uj ∪ {i + 1}) into k subsets of size k − 1, which is (k2 −k)! . ((k−1)!)k k! This implies that, at most (i − 1) (k2 − k)! ((k − 1)!)k k! different partitions of the set [k2 ] \ {i + 1} into k + 1 subsets of size k − 1, have some Uj as a partition part. Now we consider the ratio between the number of all possible partitions and the number of partitions containing some Uj as a partition part. Since k ≥ 4, (k2 −1)! ((k−1)!)k+1 (k+1)! (k2 −2)(k2 −k)! ((k−1)!)k k! = = (k2 − 3)(k2 − 4) · · · (k2 − k + 1) (k − 2)! 1 (k − 2) × (k2 − 3) (k2 − 4) (k2 − (k − 1)) × × ··· × > 1, (k − 3) (k − 4) (k − (k − 1)) and this inequality using the inequality i − 1 ≤ k2 − 2, implies that (k2 − 1)! (k2 − k)! − ( i − 1 ) > 1. ((k − 1)!)k+1 (k + 1)! ((k − 1)!)k k! Therefore, there exists a partition Ai+1,1 , Ai+1,2 , . . . , Ai+1,k+1 of the (k − 1)-subsets of [k2 ] \ {i + 1} such that none of them is equal to some Uj , 1 ≤ j ≤ i. Now, let Fi+1 := k+1  j=1 {Ai+1,j ∪ {i + 1}}. Note that, Fi+1 is starlike of size k + 1 with center i + 1, and Fi+1 ∩ Fj = ∅ for each j, 2 ≤ j ≤ i. In a similar way, we can construct Fi+2 , . . . , Fk2 . If n > k2 , then for each l, k2 + 1 ≤ l ≤ n, we define Fl := k+1  j= 1 {A2,j ∪ {l}}. From the above construction, it can be seen that, each Fj is starlike of size k + 1 with center j. Also, every two vertices in Fj have  j as their unique common symbol and hence, Fj uses exactly 1 + (k + 1)(k − 1) symbols of the set [n]. Now, if [n] \ Fj is a vertex which j ̸∈ A, then A has non-empty intersection with at most k elements of Fj for, A is a k-subset of k [n]. Thus, there exists vertex A′ ∈ Fj such that A ∩ A′ = ∅ and hence, d(A, Fj ) = 1. Therefore, properties (a) to (d) hold for A∈ this family. Now, for each l, 2 ≤ l ≤ n, we define the set F̄l as F̄l :=  i1 i2 . . . ik−1 l | i1 i2 . . . ik−1 ∈    n  [l − 1] , i1 i2 . . . ik−1 l ̸∈ Fj . k−1 j=2 Thus, each non-empty F̄l is starlike with center l. By letting Cl := Fl ∪ F̄l , 2 ≤ l ≤ n, it can be seen that C2 , C3 , . . . , Cn is a  partition of the vertex set ∏ [n] k into starlike sets and hence we have a proper (n − 1)-coloring of KG(n, k). Let = (C2 , . . . , Cn ) be an ordered partition of the vertex set into the above resulting color classes. Let i1 i2 . . . ik−1 l and i′1 i′2 . . . i′k−1 l be two distinct vertices in Cl . Since these two vertices are distinct, we can assume that i1 ̸∈ {i′1 , i′2 , . . . , i′k−1 }, i′1 ̸∈ {i1 , i2 , . . . , ik−1 }. 2219 A. Behtoei, B. Omoomi / Discrete Applied Mathematics 159 (2011) 2214–2221 Table 1 A locating 6-coloring of KG(7, 3). C1 C2 C3 C4 C5 C6 123 (0, 2, 3, 1, 1, 1) 124 (0, 2, 1, 2, 2, 1) 127 (0, 2, 1, 1, 2, 3) 134 (0, 1, 2, 2, 2, 1) 137 (0, 1, 2, 1, 1, 3) 147 (0, 1, 1, 1, 1, 3) 156 (0, 1, 1, 2, 2, 2) 157 (0, 1, 1, 2, 1, 2) 167 (0, 1, 1, 1, 2, 2) 234 (1, 0, 2, 2, 2, 1) 237 (1, 0, 2, 1, 1, 3) 247 (1, 0, 1, 1, 1, 3) 256 (1, 0, 1, 2, 2, 2) 257 (1, 0, 1, 2, 1, 2) 267 (1, 0, 1, 1, 2, 2) 345 (1, 1, 0, 2, 1, 3) 346 (1, 1, 0, 1, 2, 3) 347 (1, 1, 0, 1, 1, 3) 356 (1, 1, 0, 2, 2, 2) 357 (1, 2, 0, 2, 1, 2) 367 (1, 2, 0, 1, 2, 2) 456 (1, 1, 2, 0, 2, 2) 457 (1, 2, 2, 0, 1, 2) 125 (2, 2, 1, 0, 1, 3) 135 (2, 1, 2, 0, 1, 3) 145 (2, 1, 1, 0, 1, 3) 235 (1, 2, 2, 0, 1, 3) 245 (1, 2, 1, 0, 1, 3) 467 (1, 2, 2, 1, 0, 2) 126 (2, 2, 1, 1, 0, 3) 136 (2, 1, 2, 1, 0, 3) 146 (2, 1, 1, 1, 0, 3) 236 (1, 2, 2, 1, 0, 3) 246 (1, 2, 1, 1, 0, 3) 567 (1, 1, 2, 2, 2, 0) Without loss of generality, assume that i′1 < i1 . By the properties of the family {F2 , F3 , . . . , Fn }, it can be seen that d(i′1 i′2 . . . i′k−1 l, Ci1 ) = 1. d(i1 i2 . . . ik−1 l, Ci1 ) = 2, Hence, cΠ (i1 i2 . . . ik−1 l) ̸= cΠ (i′1 i′2 . . . i′k−1 l); accordingly, this coloring is a locating (n − 1)-coloring.  The following result gives almost tight and good upper and lower bounds for the locating chromatic number of KG(n, 3), n ≥ 7. Theorem 4. For all positive integers n ≥ 7, we have n − 4 ≤ χL (KG(n, 3)) ≤ n − 1. Proof. Since the chromatic number of the Kneser graph KG(n, 3) is n − 4, the lower bound follows. By Theorem 3, for n ≥ 9, we have χL (KG(n, 3)) ≤ n − 1. For the cases n = 7 and n = 8, we have explicit locating (n − 1)-colorings. The color classes and the color codes of the vertices are illustrated in Tables 1 and 2.  Theorems 2–4 motivate us to propose the following question. Question. Is it true that for all positive integers n, k, n > 2k, χL (KG(n, k)) = n − 1? 4. The locating chromatic number of KG (2k + 1, k ) Let k ≥ 3. Note that, χ(KG(2k + 1, k)) = 3 and by Theorem C, the diameter of KG(2k + 1, k) is k. In what follows, we present a lower bound for the locating chromatic number of odd graphs, which shows that the chromatic number and the locating chromatic number of odd graphs are far apart. Moreover, we give an upper bound for the metric dimension and accordingly for the locating chromatic number of odd graphs. Lemma 1. If k ≥ 4, then   2k+1 k ≤ kk . Proof. By a simple calculation, it can be seen that the lemma holds for k ∈ {4, 5, 6}. Obviously, for all k ≥ 7, we have +3 k2 + 5k + 6 < 2k2 . Thus, k2k × k+k 2 < 1. k+j Moreover, for each j, 4 ≤ j ≤ k + 1, we have k(j−1) < 1. Therefore,  2k+1 k kk  = = 2k + 1 k2  × 2k k2 × ··· × k+3 × k+2 −k 2k k  k+1 ∏ k+2 k+3 k+j × < 1.  × k ( j − 1 ) 2k k j=4 Theorem 5. For all positive integers k, k ≥ 3, we have dimM (KG(2k + 1, k)) ≤ 2k + 1, and  2k ln 2 − 1 8k − 21 ln(kπ ) ln k  ≤ χL (KG(2k + 1, k)) ≤ 2k + 4. Particularly, χ(KG(2k + 1, k)) < χL (KG(2k + 1, k)). 2220 C1 C2 C3 C4 C5 C6 C7 128 (0, 2, 1, 1, 1, 1, 1) 138 (0, 1, 2, 1, 1, 2, 2) 145 (0, 1, 1, 2, 2, 1, 1) 156 (0, 1, 1, 1, 2, 2, 1) 168 (0, 1, 1, 1, 1, 2, 1) 178 (0, 1, 1, 1, 2, 1, 2) 234 (1, 0, 2, 2, 1, 1, 1) 235 (1, 0, 2, 1, 2, 1, 1) 238 (1, 0, 2, 1, 1, 1, 1) 245 (1, 0, 1, 2, 2, 1, 1) 247 (1, 0, 1, 2, 1, 1, 2) 256 (1, 0, 1, 1, 2, 2, 1) 258 (2, 0, 1, 1, 2, 1, 1) 267 (1, 0, 1, 1, 1, 2, 2) 123 (2, 2, 0, 1, 1, 1, 1) 236 (1, 2, 0, 1, 1, 2, 1) 237 (1, 2, 0, 1, 1, 1, 2) 345 (1, 1, 0, 2, 2, 1, 1) 348 (1, 1, 0, 2, 1, 1, 1) 356 (1, 1, 0, 1, 2, 2, 1) 358 (2, 1, 0, 1, 2, 1, 1) 367 (1, 1, 0, 1, 1, 2, 2) 368 (1, 1, 0, 1, 1, 2, 1) 124 (2, 2, 1, 0, 1, 1, 1) 134 (2, 1, 2, 0, 1, 1, 1) 148 (2, 1, 1, 0, 1, 2, 2) 246 (1, 2, 1, 0, 1, 2, 1) 248 (1, 2, 1, 0, 1, 1, 1) 456 (1, 1, 1, 0, 2, 2, 1) 457 (1, 1, 1, 0, 2, 1, 2) 467 (1, 1, 1, 0, 1, 2, 2) 468 (2, 1, 1, 0, 1, 2, 1) 125 (2, 2, 1, 1, 0, 1, 1) 135 (2, 1, 2, 1, 0, 1, 1) 157 (2, 1, 1, 1, 0, 1, 2) 158 (2, 1, 1, 1, 0, 1, 1) 257 (1, 2, 1, 1, 0, 1, 2) 357 (1, 2, 2, 1, 0, 1, 2) 458 (2, 1, 1, 2, 0, 1, 1) 567 (1, 1, 1, 1, 0, 2, 2) 126 (2, 2, 1, 1, 1, 0, 1) 136 (2, 1, 2, 1, 1, 0, 1) 146 (2, 1, 1, 2, 1, 0, 1) 167 (2, 1, 1, 1, 1, 0, 2) 268 (1, 2, 1, 1, 1, 0, 1) 346 (1, 1, 2, 2, 1, 0, 1) 568 (2, 1, 1, 1, 2, 0, 1) 678 (1, 1, 1, 1, 1, 0, 2) 127 (2, 2, 1, 1, 1, 1, 0) 137 (2, 1, 2, 1, 1, 1, 0) 147 (2, 1, 1, 2, 2, 1, 0) 278 (1, 2, 1, 1, 1, 1, 0) 347 (1, 1, 2, 2, 1, 1, 0) 378 (1, 1, 2, 1, 1, 1, 0) 478 (1, 1, 1, 2, 1, 1, 0) 578 (2, 1, 1, 1, 2, 1, 0) A. Behtoei, B. Omoomi / Discrete Applied Mathematics 159 (2011) 2214–2221 Table 2 A locating 7-coloring of KG(8, 3). A. Behtoei, B. Omoomi / Discrete Applied Mathematics 159 (2011) 2214–2221 2221 Proof. By Theorem C, if A, B are two distinct vertices in KG(2k + 1, k), then d(A, B) = min{2(k − |A ∩ B|), 2|A ∩ B| + 1}. This implies that the possible values of d(A, B) are in one-to-one correspondence with the possible values {0, 1, 2, . . . , k − 1} of |A ∩ B|. Thus, we have d(A, B) = d(A1 , B1 ) if and only if |A ∩ B| = |A1 ∩ B1 |, where A1 , B1 ∈ V (KG(2k + 1, k)). For each i, 1 ≤ i ≤ 2k + 1, let wi := {i, i + 1, . . . , i + (k − 1)}, where the elements are considered modulo 2k + 1. For example, w1 = {1, 2, . . . , k} and w2k+1 = {2k + 1, 1, 2, . . . , k − 1}. We want to show that for each two distinct vertices A, B, there exists a vertex wi , 1 ≤ i ≤ 2k + 1, such that d(A, wi ) ̸= d(B, wi ), equivalently |A ∩ wi | ̸= |B ∩ wi |. Suppose that, on the contrary, there exist vertices A and B such that |A ∩ wi | = |B ∩ wi | for all i ∈ {1, 2, . . . , 2k + 1}. Let A′ := A \ B and B′ := B \ A. Thus, we have 1 ≤ |A′ | = |B′ | = k − |A ∩ B| ≤ k, and |A′ ∩ wi | = |B′ ∩ wi | for each i ∈ {1, 2, . . . , 2k + 1}. Let a ∈ A′ . Since A′ ∩ B′ = ∅, a ̸∈ B′ and |A′ ∩ wa | = |B′ ∩ wa | ≤ |B′ ∩ wa+1 | = |A′ ∩ wa+1 |. Moreover, a ∈ A′ ∩ wa , a ̸∈ A′ ∩ wa+1 and wa+1 \ wa = {a + k}. Therefore, a + k ∈ A′ . Similarly, a + jk ∈ A′ (mod 2k + 1), for each j ∈ N. If a + jk ≡ a + j′ k (mod 2k + 1), then j ≡ j′ (mod 2k + 1). This implies that |A′ | = 2k + 1, which is a contradiction. Hence, the set {w1 , . . . , w2k+1 } is a resolving set, and dimM (KG(2k + 1, k)) ≤ 2k + 1. By Theorem B, we have χL (KG(2k + 1, k)) ≤ dim(KG(2k + 1, k)) + χ(KG(2k + 1, k)) ≤ 2k + 4. M Finally, we show that the lower bound holds. Using Theorem 1, it is easy to see that the inequality holds for k = 3. 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