2-tone Colorings in Graph Products

Transcription

1 -tone Colorings in Graph Products Jennifer Loe, Danielle Middlebrooks, Ashley Morris August 13, 013 Abstract A variation of graph coloring known as a t-tone k-coloring assigns a set of t colors to each vertex of a graph from the set 1,, k where the sets of colors assigned to any two vertices distance d apart share fewer than d colors in common. The minimum integer k such that a graph G has a t-tone k-coloring is known as the t-tone chromatic number. We study the -tone chromatic number in three different graph products. In particular, given graphs G and H, we bound the -tone chromatic number for the direct product G H, the Cartesian product G H, and the strong product G H. 1 Introduction Many variations of classic graph k-colorings abound, whereby we take a k-coloring of a graph to mean an assignment of an element from 1,, k, called a color, to each of the vertices of the graph. Gary Chartrand was the first to introduce a t-tone k-coloring, which is an assignment of t elements from the set 1,, k to each vertex such that the sets of colors assigned to any two distinct vertices within distance d share fewer than d colors. This t-tone k-coloring variation can be viewed as a generalization of classic graph coloring since a 1-tone k-coloring of a graph is simply a k-coloring. In an initial investigation of this graph parameter, Fonger, Goss, Phillips, and Segroves [4] determined the exact value for the -tone chromatic number of a tree and a complete multipartite graph. In 011, Bickle and Phillips [] gave general bounds for the t-tone chromatic number of a graph G in terms of the maximum degree of G. Cranston, Kim, and Kinnersley [3] improved upon these bounds shortly thereafter. In [1], the problem of determining bounds of the t-tone chromatic number of the Cartesian product of two graphs is posed. In this paper, we determine bounds on the -tone chromatic number in the direct product, Cartesian product, and the strong product of two graphs. 1

2 1.1 Definitions and Notation An undirected graph G = (V, E) is a set of vertices V (G) together with a set of edges E(G) which are unordered pairs (x, y), or simply xy, of vertices of G. If xy E(G), we say that x and y are adjacent. For the purpose of this paper, we consider only simple undirected graphs, meaning undirected graphs that have no loops. For any vertex x V (G), the open neighborhood of x, denoted N(x), is defined to be the set of all vertices adjacent to x. The closed neighborhood of x is defined to be N[x] = N(x) x. The degree of x, denoted deg G (x), is the cardinality of the open neighborhood of x. We let (G) = max x V (G) deg G (x). A path of length k, denoted P k, in G between two vertices v 0 and v k is a sequence v 0,, v k of k + 1 distinct vertices of G such that for all i 0,, k 1, the edge v i v i+1 E(G). The distance between vertices u and v, denoted d G (u, v) is the smallest length path between u and v. When the context is clear, we use the shorthand notation d(u, v). As stated before, a proper k-coloring of a graph G is an assignment of an element from 1,, k, called a color, to each vertex in V (G) such that no two adjacent vertices are assigned the same color. The chromatic number of G, denoted χ(g), is the minimum number k such that G has a proper k-coloring. We use K n to denote the complete graph on n vertices, which is a graph with n vertices where each vertex has degree n 1. Given a graph G, a clique is any complete subgraph of G, and the clique number of G, denoted ω(g), is the cardinality of the maximum clique of G. For positive integers t and k where t k, we let [k] represent the set 1,, k and denote the family of t-element subsets of [k] by ( ) [k] t. We now give a formal definition of a t-tone k-coloring of a graph. Definition 1. Let G be a graph and let t and k be positive integers such that t k. A t-tone k-coloring of G is a function f : V (G) ( ) [k] t such that f(u) f(v) < dg (u, v) for all distinct vertices u and v. A graph that has a t-tone k-coloring is said to be t-tone k-colorable. The t-tone chromatic number of G, denoted τ t (G), is the minimum k such that G is t-tone k-colorable. Given a t-tone k-coloring f of G, we call f(v) the label of v and the elements of [k] colors. Figure 1 depicts a -tone 5-coloring of P 5 which is, indeed, minimum. 1, 3,4 1,5,4 1,3 Figure 1: A -tone 5-coloring of P 5 In this paper, we focus on -tone k-colorings of three different graph products. First, we consider the direct product. Definition. Given two graphs G and H, the direct product of G and H, denoted G H, is the graph whose vertex set is the Cartesian product V (G) V (H) and whose edge set is E(G H) = (x 1, y 1 )(x, y ) x 1 x E(G) and y 1 y E(H).

3 Figure depicts the direct product of K and K 3. Figure : K K 3 The next graph we consider is the Cartesian product, which is the motivation behind this research. We aim to improve upon the upper bound for the -tone chromatic number of the Cartesian product of two graphs given in [1]. Definition 3. The Cartesian product of graphs G and H, denoted G H, is the graph whose vertex set is V (G) V (H) with edge set E(G H) = (x 1, y 1 )(x, y ) x 1 y 1 E(G) and x = y, or x 1 = y 1 and x y E(H). Figure 3 depicts the Cartesian product of K and K 3. Figure 3: K K 3 The last product we consider is the strong product of two graphs. Definition 4. The strong product of graphs G and H, denoted G H, is the graph whose vertex set is the Cartesian product V (G) V (H) and whose edge set is E(G H) = E(G H) E(G H). Figure 4 depicts the strong product of K and K 3. Previous Results In this section, we state various results that were useful in our research. We give a short proof of any result that will be used in Section 3. The following theorem shows the monotonicity of t-tone colorings. 3

4 Figure 4: K K 3 Theorem 5. [4] If H is a subgraph of G, then τ t (H) τ t (G). Proof. Let f be a proper t-tone τ t (G)-coloring of G. The restriction of f to H must then be a proper t-tone coloring of H. Thus, τ t (H) is no greater than τ t (G). In Section 3, we determine the -tone chromatic number of K m K n for any m, n Z. We shall use the following result to do so. Theorem 6. [4] If G = K n for any positive integer n, then τ t (K n ) = tn. Proof. Notice that each pair of distinct vertices of V (K n ) is adjacent. Therefore, the set of t-element labels of the n vertices of K n must be pairwise disjoint, and we have τ t (K n ) = tn. Recall that a k-partite graph is a graph whose vertices can be partitioned into k disjoint sets, called partite sets, such that no two vertices within the same partite set are adjacent. A complete k-partite graph, denoted K a1,,a k, is a k-partite graph with the additional property that any two distinct vertices contained in different partite sets are adjacent, and whose partite sets are of orders a 1,, a k. Theorem 7. [4] Let G = K a1,a,...,a k denote the complete k-partite graph, with partite sets of orders a 1, a,..., a k. Then k 1 + 8ai + 1 τ (G) =. i=1 Proof. Let G = K a1,a,...,a k and let χ(g) = k. Let A 1,..., A k denote the color classes of G. Let A i = a i, for 1 i k. Now let f be a proper -tone τ (G)-coloring of G. Notice that for any pair of distinct vertices u, v A i, we have d(u, v) =. Hence, f(v) f(u) 1. So the number of colors required for the i th partite set is s i such that ( s i ) ai. Thus, we get that s i = 1+ 8ai +1. Furthermore, since every element in A i is adjacent to every element in A j for all i j with 1 i, j k, they cannot have any colors in common. Therefore, the -tone chromatic number of G is the sum of all such s i, as desired. There is a natural relationship between t-tone colorings and another variation of graph coloring known as a distance (d, k)-coloring. 4

5 Definition 8. A distance (d, k)-coloring of G is a mapping f : V (G) [k] such that if u and v are two distinct vertices of V (G) where d(u, v) d, then f(u) f(v). We use χ d (G) to denote the minimum number k such that G has a proper distance (d, k)-coloring. As stated throughout the literature, determining χ d (G) is equivalent to determining the chromatic number of the d th power of a graph, defined below. Definition 9. Given a graph G, the d th power of G, denoted G d,is defined to be the graph with V (G d ) = V (G), and for any two distinct vertices u, v V (G d ), uv E(G d ) if and only if d G (u, v) d. Since the focus of this paper is on -tone colorings, we will restrict ourselves to distance (, k)- colorings. We now prove equality between χ (G) and χ(g ). Theorem 10. For any graph G, χ (G) = χ(g ). Proof. Assume that f : V (G) 1,..., χ (G) is a proper distance (, χ (G))-coloring of G. If uv E(G ), then d G (u, v) so f(u) f(v). Thus, f is a proper χ (G)-coloring of G, and we have χ(g ) χ (G). In the other direction, assume that g : V (G ) 1,, χ(g ) is a proper χ(g )-coloring of G. If d G (u, v), then uv E(G ) so g(u) g(v). Thus, g is a distance (, χ(g ))-coloring of G, and the result follows. As noted in Fonger, Phillips and Segroves [4], τ (G) is related to both χ(g) and χ(g ) in the following way. Theorem 11. [4] For any graph G, τ (G) χ(g) + χ(g ). Proof. Let f 1 be a chromatic coloring of G using the colors 1,,..., χ(g), and f a chromatic coloring of G using the colors χ(g) + 1,..., χ(g ). Then we define a -tone coloring f of G by setting f(v) = f 1 (v), f (v). Then, if u, v are adjacent in G, we have that f 1 (v), f (v), f 1 (u), and f (u) are four different colors. Similarly, if d(u, v) = for vertices u and v of G, then f (u) f (v), and so the labels on u and v assigned by f are not the same. Therefore, f is a proper -tone coloring of G using χ(g) + χ(g ) colors. Bickle and Phillips give an upper bound for τ (G) in [] based on (G). Theorem 1. [] For any graph G, τ (G) [ (G)] + (G). This bound was improved upon by Cranston, Kim and Kinnersley in [3]. We will reference this result throughout Section 3 as a comparison to our results. Theorem 13. [3] For any graph G, τ (G) ( + ) (G). 5

6 3 New Results 3.1 Direct Product In this section, we focus on the direct product of two graphs. We restate the definition of the direct product for ease of reference. Definition 14. Given two graphs G and H, the direct product of G and H, denoted G H, is the graph whose vertex set is the Cartesian product V (G) V (H) and whose edge set is E(G H) = (x 1, y 1 )(x, y ) x 1 x E(G) and y 1 y E(H). Throughout this section, when we consider the direct product of two graphs G and H where G = m and H = n, we will represent the vertices of G as x 1,, x m and the vertices of H as y 1,, y n. Using this notation, for each i [m] we define a column C i as the set of all vertices with first coordinate x i. Similarly, for each j [n] we define the row R j as the set of all vertices with second coordinate y j. In particular, for i [m], the i th column is C i = (i, j) j [n]. Similarly, for j [n] the j th row is the set R j = (i, j) i [m]. In order to find an upper and lower bound of the -tone chromatic number of the direct product of any two graphs G and H, we first considered the direct product of two complete graphs. The following property follows from the definition of the direct product. Property 15. For any m, n Z V (K m K n ) = (x i, y k ) 1 i m and 1 k n, and E(K m K n ) = (x i, y k )(x j, y l ) i j and k l. Note the following consequence of Property 15. Proposition 16. Let m and n be positive integers such that m and n 3. For any two distinct vertices (x i, y j ) and (x i, y k ) of V (K m K n ), d((x i, y j ), (x i, y k )) =. Proof. Let u, v V (K m K n ) be two distinct vertices, and write u = (x i, y j ) and v = (x i, y k ) for some 1 i m and 1 j < k n. Note that u and v are both contained in column C i. Since m 3, there exists a 1,, m such that a i. Thus, x i x a E(K m ). Moreover, since n 3, there exists b 1,, n such that b j and b k. Therefore, y j y b E(H) and y b y k E(H). So (x i, y j )(x a, y b )(x i, y k ) is a path in K m K n of length. It follows that d(u, v) =. Recall from Section 1 that given a graph G and a t-tone k-coloring f of G, we call f(v) the label of v and the elements of [k] colors. Additionally, for any set of vertices A V (G), we define the set of colors contained in the labels associated with A to be c(a) = c [k] c f(v) for some v A. 6

7 Theorem 17. If m, n N where m n and t = n, then τ (K m K n ) min m t, m + n, m t + n t ( t 1) Proof. First consider the case where m = n =. Since K K = K, τ (K K ) = τ (K ) = 4. One can easily verify that this is the minimum of the three functions. Now consider all other cases where m, n m and n. Let f be a minimum -tone k-coloring of K m K n. For each 1 i m, define A i as the set of all vertices v C i such that for each a f(v), there exists a vertex w C i with w v and a f(v) f(w). Note the following consequence of this partition of each C i. Fix i 1,, m and let v and w be distinct vertices of A i as described above. That is, for some a f(v) we have a f(v) f(w). This implies that for all j 1,, m such that j i and for any u A j, a f(u) since u is adjacent to at least one of v or w by Property 15. Therefore, the set of colors contained in the labels associated with A i is disjoint from the set of colors contained in the labels associated with A j. That is, c(a i ) c(a j ) =. For each 1 i m, let s i = c(a i ). We may assume that min 1 i m s i = s l for some l 1,, m. Thus, the number of distinct colors contained in the labels associated with m i=1a i is at least ms l. Furthermore, for each (x l, y j ) C l \A l, there exists a color a f((x l, y j )) such that a is not contained in any other label associated with C l. So for each 1 i m, a c(a i ) by definition of A i. Since f is a proper -tone k-coloring of K m K n, we know that k ms l + C l \A l. We now determine the minimum k based on the value of C l \A l. Case 1: Assume that C l \A l = n. Thus, A l = and s l = 0. Let j represent the number of columns such that s i > 0 for some i 1,, m. Since s l = 0, m j 1 or equivalently m j + 1. Let A = C i i 1,, m and s i = 0. For indexing purposes, we shall write A = C α(1),, C α(m j) where α(i) 1,, m for 1 i m j. Since m j m n, there exists a set W = v α(1),, v α(m j) such that v α(i) C α(i) for each α(i) and if α(i) α(j), then v α(i) and v α(j) are contained in different rows. This implies that the induced subgraph of W is a clique, and it follows that f(v α(i) ) f(v α(j) ) = 0 when α(i) α(j). Furthermore, the n (m j) remaining vertices (x l, y j ) C l \A l where R j W = contain at least n (m j) colors that are not contained in c(w ). Thus, c(b) (m j) + n (m j). Finally, for each column C i where s i > 0, we know that s i. Thus, k (m j) + n (m j) + j = m + n + j m + n.. Case : Assume that C l \A l = 0. It follows that A l = C l and A l = n. Furthermore, since s l represents the number of distinct colors contained in the labels associated with A l, we know 7

8 that s l. Using the same argument as in Theorem 7 and the fact that any two distinct vertices u, v A l satisfy d(u, v) =, we know that ( s l ) n. Using the quadratic formula, this implies that s l. Consequently, k m n n Case 3: Assume that n > C l \A l > 0. If ( s l ) > n, then clearly k m n ( sl ) n, or equivalently 1+ sl 1+8n implies C l \A l = n A l. As in Case 1, we know ( s l ) Al. Thus, ( ) sl n n A l ( ) sl = n C l \A l. Therefore, ms l + C l \A l ms l + n. So assume. We have n = C l = A l + C l \A l, which ( ) sl = ms l + n s l(s l 1). So we consider the function g(s) = ms + n s(s 1) 1+ over the interval s 1+8n. One can easily verify that g (s) = m s + 1 and g (s) = 1. Thus, g is concave down 1+ for all values of s 1+8n, and over this interval g has a local maximum when s = m + 1. Therefore, the local minimums for g occur when s = and s =. Letting t = n, it follows that k ms l + C l \A l s(s 1) min ms + n s n s.t. s t ( t 1) min m + n 1, m t + n n Since m + n 1 > m + n and f is assumed to be a minimum -tone coloring, we shall take min m + n 1, m t + n t ( t 1) = m t + n t ( t 1). Note that these cases sometimes overlap. For example, ( s ) = n implies that t = t = t and n t ( t 1) = 0, resulting in m t = m t + n t ( t 1). In any case, we have t ( t 1) τ (K m K n ) min m t, m + n, m t + n. 8

9 Theorem 18. If m, n N where m n and t = n, then τ (K m K n ) = min m t, m + n, m t + n t ( t 1) Proof. From Theorem 17, all that remains to be shown is that t ( t 1) τ (K m K n ) min m t, m + n, m t + n. Given m, n N where m n and t = n, we shall construct different -tone colorings which depend on the value of min m t, m + n, m t + n t ( t 1). Case 1: First assume that min m t, m + n, m t + n t ( t 1) = m t. Choose m pairwise disjoint sets each containing t distinct colors, and denote each set of colors S i for 1 i m. Since ( ) t n, for each 1 i m there exist n distinct combinations containing two colors from the set S i. Thus, we may define f : V (K m K n ) ( ) [ t ] to be any mapping such that for each 1 i m the restriction of f to the set of vertices in C i is an injective mapping to the set of combinations containing two colors from the set S i. Figure 5 illustrates this particular -tone coloring for K K 6. To see that f is a proper -tone coloring of K m K n, let u and v be distinct vertices of V (K m K n ). If u and v are not contained in the same column, then f(u) f(v) =. So assume u, v C i for some i 1,, m. Since d(u, v) =, we must show that f(u) f(v) 1. However, this follows from the fact that f does not assign any label to more than one vertex of C i. Therefore, f is a proper -tone coloring.. 3,4 7,8,4 6,8 1,4 5,8 1,3 5,7,3 6,7 1, 5,6 Figure 5: K K 6 9

10 Case : Next, assume that min m t, m + n, m t + n t ( t 1) = m + n. Let f 1 be a proper coloring of K m and f be a proper coloring of K n defined as follows: f 1 : V (K m ) [m] x i i f : V (K n ) m + 1,, m + n y k k + m. Define the following function on V (K m K n ): ( ) [m + n] g : V (K m K n ) (x i, y k ) f 1 (x i ), f (y k ). Figure 6 illustrates a particular -tone coloring of this type for K K 3. 1,4 4,5 1,3 3,5 1,,5 Figure 6: K K 3 We claim g is a proper -tone coloring of K m K n. Clearly g((x, y)) = for all (x, y) V (K m K n ). Let (x i, y k ) and (x j, y l ) be two distinct vertices of V (K m K n ) where 1 i, j m and 1 k, l n. Then g((x i, y k )) = i, k + m and g((x j, y l )) = j, l + m. If (x i, y k ) and (x j, y l ) are adjacent, then i j and k l. Thus, g((x i, y k )) g((x j, y l )) = 0. If d((x i, y k ), (x j, y l )) =, then either i j or k l. In any case, g((x i, y k )) g((x j, y l )) 1. Therefore, g is a proper -tone coloring of K m K n, and we may conclude that τ (K m K n ) m + n. Case 3: Assume that min m t, m + n, m t + n t ( t 1) = m t + n t ( t 1). Note that t = n is the only positive solution to ( ) ( t = n. Therefore, t satisfies t ) ( n. Let s = t ) and consider the subgraph H of K m K n induced by the set (x i, y j ) 1 i m, 1 j s. Thus, H = K m K s. As in Case, choose m pairwise disjoint sets of t distinct colors and denote each set S i for 1 i m. Define f 1 : V (H) ( ) [ t ] to be any mapping such that for 10

11 each 1 i m, the restriction of f 1 to the set of vertices of C i is an injective mapping to the set of combinations containing two colors from the set S i. A similar argument as in Case can be used to show that f 1 is a proper -tone coloring of H. Next, choose n s distinct colors each of which are not contained in the set m i=1s i, and label these colors t s+1,, t n. Additionally, for each i 1,, m, choose one color from the set S i and call it c i. Notice that V (K m K n )\V (H) = (x i, y j ) 1 i m, s + 1 j n. Define ( ) [m + n s] f : V (K m K n )\V (H) (x i, y j ) c i, t j. We claim that f is a proper -tone coloring of (K m K n )\H. To see this, let (x i, y k ) and (x j, y l ) be two distinct vertices of V (K m K n )\V (H) for some 1 i, j m and s + 1 k, l n. If (x i, y k ) and (x j, y l ) are adjacent, then i j and k l. Since S i and S j are two disjoint sets of colors, we know that c i c j. Moreover, we know that t k t l since k l. Thus, f ((x i, y k )) f ((x j, y l )) = 0. If d((x i, y k ), (x j, y l )) =, then either i j or k l. It follows that f ((x i, y k )) f ((x j, y l )) 1. Therefore, f is a proper -tone coloring of (K m K n )\H. ) such that Now define g : V (K m K n ) ( [m t +n s] f 1 (u) g(u) = f (u) if u V (H) otherwise. Figure 7 illustrates a particular -tone coloring of this type for K K 11. To see that g is a proper -tone coloring, we only need to consider when u V (H) and v V (H). Write u = (x i, y k ) and v = (x j, y l ) for some i, j 1,, m, k 1,, s, and l s + 1,, n. By definition g(v) = c j, t l, and we know t l g(u) since u V (H). So if u and v are located in the same column, then g(u) g(v) 1. If u and v are not located in the same column, then i j and c j g(u) since c j S i. It follows that g(u) g(v) = 0. Therefore, g is a proper -tone coloring of K m K n using m t + n ( ) t colors. Using similar ideas found in Theorems 17-18, we can bound the value of τ (G H) given any graphs G and H. We make use of the following general lower bound given in [4]. Theorem 19. [4] Let G be a graph and let (G) = d. Then 8d τ (G). 11

12 1,11 4,5 3,5 3,4,5,4,3 1,5 1,4 1,3 1, 6,11 9,10 8,10 8,9 7,10 7,9 7,8 6,10 6,9 6,8 6,7 Figure 7: K K 11 Theorem 0. Given two graphs G and H, (G) (H) max, τ (K ω(g) K ω(h) ) τ (G H) χ(g ) + χ(h ). Proof. We first show that for any graphs G and H, we have τ (G H) χ(g ) + χ(h ). Assume χ (G) = k 1 and χ (H) = k. Let f 1 : V (G) 1,..., k 1 be a distance- coloring of G, and let f : V (H) k 1 + 1,..., k 1 + k be a distance- coloring of H. Define ( ) [k1 + k ] g : V (G H) such that (x, y) f 1 (x), f (y) for all x V (G) and y V (H). We claim that g is a proper -tone coloring of G H. Clearly g((x, y)) = for all (x, y) V (G H). Let (u, v) and (w, z) be two distinct vertices of V (G H). If (u, v) and (w, z) are adjacent, then uw E(G) and vz E(H). It follows that f 1 (u) f 1 (w) and f (v) f (z). Since the range of f 1 is disjoint from the range of f as mappings, we have g((u, v)) g((w, z)) = 0. If d G H ((u, v), (w, z)) =, then there exists a vertex (x, y) V (G H) such that uxw is a path in G and vyz is a path in H. Since d G (u, w) and d H (v, z), we know that f 1 (u) f 1 (w) and f (v) f (z). Thus, g((u, v)) g((w, z)) = 0, and we may conclude that g is a proper -tone coloring of G H. 1

13 (G) (H) Next, we show that max, τ (K ω(g) K ω(h) ) τ (G H). We shall assume that ω(g) is the clique number of G, and ω(h) is the clique number of H. By definition of the direct product, K ω(g) K ω(h) is a subgraph of G H. Thus, τ (K ω(g) K ω(h) ) τ (G H). On the other hand, we know (G H) = (G) (H). So by Theorem 19, we know that (G) (H) τ (G H). Therefore, max (G) (H), τ (K ω(g) K ω(h) ) τ (G H). It should be noted that there exist graphs G and H such that the upper bound in Theorem 0 is better than applying Theorem 13. For example, consider the graph P 3 P 4 in Figure 8. One can easily verify that the labeling shown in Figure 8 is in fact a -tone coloring. Thus, τ (P 3 P 4 ) 5 which is an improvement from the bound given in Theorem 13 of τ (P 3 P 4 ) ( + ()) (P 3 P 4 ) = ( + )4 = ,4,4 3,4 6 1,6,6 3,6 5 1,5,5 3,5 4 1,4,4 3,4 1 3 Figure 8: A -tone coloring of P 3 P 4 13

14 3. Cartesian Product We now focus on the Cartesian product of two graphs. We restate the definition of this product below for ease of reference. Definition 1. The Cartesian product of graphs G and H, denoted G H, is the graph whose vertex set is V (G) V (H) with edge set E(G H) = (x 1, y 1 )(x, y ) x 1 y 1 E(G) and x = y, or x 1 = y 1 and x y E(H). In this particular product, we have an obvious lower bound for the -tone chromatic number of G H. Theorem. Given two graphs G and H, maxτ (G), τ (H) τ (G H). Proof. This follows from the fact that G and H are both subgraphs of G H. In terms of an upper bound, it is stated in [1] that τ (G H) τ (G)τ (H), but that this bound could be improved. We give an upper bound for τ (G H) in terms of maxχ(g ), χ(h ) depending on the parity of this value. Theorem 3. Given two graphs G and H where maxχ(g ), χ(h ) = χ(g ), χ(g ) if χ(g ) is odd τ (G H) (χ(g ) + 1) otherwise. Proof. Assume that maxχ(g ), χ(h ) = χ(g ). If χ(g ) is an even integer, then we will let k = χ(g ) + 1. Otherwise, we will let k = χ(g ). Let f 1 : V (G) [k] be a proper distance (, k)-coloring of G, and let f : V (H) [k] be a proper distance (, k)-coloring of H. Define g : V (G H) ( ) [k] such that (x, y) f 1 (x) + f (y) (mod k), (f (y) f 1 (x) (mod k)) + k. We will first show that g assigns two distinct colors to each vertex of G H. Let (x, y) V (G H) and write g((x, y)) = a, b. Since a = f 1 (x) + f (y) (mod k), it follows that a [k]. On the other hand, b = (f (y) f 1 (x) (mod k)) + k so b k + 1,, k. Thus, g((x, y)) =. Next, we show that g satisfies the distance criteria for -tone colorings. Let (u, v) and (w, z) be two distinct vertices of V (G H). 14

15 Case 1: Suppose that d G H ((u, v)(w, z)) = 1. Then either u = w and d H (v, z) = 1 or v = z and d G (u, w) = 1. If u = w and d H (v, z) = 1, then we know that f 1 (u) = f 1 (w) and f (v) f (z). This implies that f 1 (u) + f (v) f 1 (w) + f (z) (mod k). Moreover, f (v) f 1 (u) f (z) f 1 (w) (mod k) = (f (v) f 1 (u) (mod k)) + k (f (z) f 1 (w) (mod k)) + k. So g((u, v)) g((w, z)) = 0. A similar argument shows that g((u, v)) g((w, z)) = 0 if v = z and d G (u, w) = 1. Case : Suppose that d G H ((u, v)(w, z)) =. Then exactly one of the following will be true. (a) (b) u = w and d H (v, z) =, or v = z and d G (u, w) =, or (c) d G (u, w) = 1 and d H (v, z) = 1. In the case of either (a) or (b), a similar argument as in Case 1 shows g((u, v)) g((w, z)) = 0. So assume d G (u, w) = 1 and d H (v, z) = 1. It follows that f 1 (u) f 1 (w) and f (v) f (z). If g((u, v)) g((w, z)) 1, we are done. So suppose that g((u, v)) = g((w, z)). Thus, Moreover, (f (v) f 1 (u) (mod k)) + k = (f (z) f 1 (w) (mod k)) + k = f (v) f 1 (u) f (z) f 1 (w) (mod k) = f (v) f (z) f 1 (u) f 1 (w) (mod k). (1) f 1 (u) + f (v) f 1 (w) + f (z) (mod k) = f 1 (u) f 1 (w) f (z) f (v) (mod k) = f (v) f 1 (z) f (z) f (v) (mod k) from (1) = f (v) f (z) (mod k). However, this cannot happen since f (v) f (z) (mod k) and gcd(, k) = 1. Thus, g((u, v)) g((w, z)) 1. Although the upper bound in Theorem 3 does not involve τ (G) or τ (H), it should be noted that there are graphs for which the upper bound is best possible. For example, consider the graph P 3 P 3. We know τ (P 3 ) = 5 and χ(p 3 ) = 3. Moreover, P 3 P 3 contains a cycle of length 4 and since τ (C 4 ) = 6, it must be the case that 6 τ (P 3 P 3 ). Figure 9 shows a proper -tone coloring using only 6 colors. Thus, τ (P 3 P 3 ) = χ(g ) = 6. 15

16 3,5 3,4 1,6 1,4,6 3,5 1 3,6 1,5,4 1 3 Figure 9: A -tone coloring of P 3 P Strong Product The last graph product that we consider is the strong product G H. We restate the definition of this product below for ease of reference. Definition 4. The strong product of graphs G and H, denoted G H, is the graph whose vertex set is the Cartesian product V (G) V (H) and whose edge set is E(G H) = E(G H) E(G H). Using similar ideas to those found in Sections 3.1 and 3., we have the following upper and lower bounds for τ (G H). Theorem 5. Given two graphs G and H, maxτ (G H), τ (G H) τ (G H) minτ (G)χ(H ), χ(g )τ (H). Proof. Note that G H and G H are both subgraphs of G H. Thus, maxτ (G H), τ (G H) τ (G H) by Theorem 5. Next, we will prove that τ (G H) minτ (G)χ(H ), χ(g )τ (H). Without loss of generality, we may assume τ (G)χ(H ) χ(g )τ (H). Let f 1 be a proper -tone coloring of G using the colors 1,,...τ (G). Let f be a proper distance (, k)-coloring of H using the colors 1, τ (G) + 1, τ (G) + 1,..., (k 1)τ (G) + 1 where k = χ(h ). Define g : V (G H) ( [kτ ) (G)+1] such that for each (x, y) V (G H) and for each c f 1 (x), we have c + f (y) g((x, y)). We show that g is a proper -tone coloring of G H. V (G H). 16 Let (u, v) and (w, z) be vertices of

17 Case 1: Assume that d G H ((u, v), (w, z)) = 1. By definition of the strong product, exactly one of the following will be true. 1) d G (u, w) = 1 and v = z, ) u = w and d H (v, z) = 1, or 3) d G (u, w) = 1 and d H (v, z) = 1. We show that g((u, v)) g((w, z)) = 0 in each of the above cases. 1) Assume d G (u, w) = 1 and v = z. Since f 1 is a proper -tone coloring of G, f 1 (u) f 1 (w) =. Thus, we can write f 1 (u) = c 1, c and f 1 (w) = c 3, c 4 where c i c j for 1 i < j 4. Since f (v) = f (z), we know for i 1, and j 3, 4 that c i + f (v) c j + f (z). Therefore, g((u, v)) g((w, z)) = 0. ) Assume u = w and d H (v, z) = 1. Since f is a proper distance (, k)-coloring of H, f (v) f (z) and we may write f (v) = iτ (G) + 1 and f (z) = jτ (G) + 1 for some 0 i < j k 1. Let f 1 (u) = c 1, c where c 1 c. Thus, g((u, v)) = c 1 + iτ (G) + 1, c + iτ (G) + 1 and g((w, z)) = c 1 + jτ (G) + 1, c + jτ (G) + 1. It is clear that c 1 + iτ (G) + 1 c 1 + jτ (G) + 1 since i < j. Similarly, c + iτ (G) + 1 c + jτ (G) + 1. Note that if c 1 + iτ (G) + 1 = c + jτ (G) + 1, then c 1 c = (j 1)τ (G). We know that c 1 c 0 since i < j. On the other hand, c 1 c cannot be a multiple of τ (G) since 1 c 1, c τ (G). Therefore, and a similar argument shows that Thus, g((u, v)) g((w, z)) = 0. c 1 + iτ (G) + 1 c + jτ (G) + 1, c + iτ (G) + 1 c 1 + jτ (G) ) Assume d G (u, w) = 1 and d H (v, z) = 1. It follows that f 1 (u) f 1 (w) = and f (v) f (z). As before, let f 1 (u) = c 1, c and f 1 (w) = c 3, c 4 where c a c b when 1 a < b 4. Also, write f (v) = iτ (G) + 1 and f (z) = jτ (G) + 1 for some 0 i < j k 1. Thus, g((u, v)) = c 1 + iτ (G) + 1, c + iτ (G) + 1 and g((w, z)) = c 3 + jτ (G) + 1, c 4 + jτ (G)

18 Again, we see that c 1 +iτ (G)+1 c +iτ (G)+1 since c 1 c. Similarly, c 3 +jτ (G)+1 c 4 + jτ (G) + 1 since c 3 c 4. Furthermore, for any a 1, and b 3, 4, we know c a + iτ (G) + 1 c b + jτ (G) + 1 since c a c b cannot be a multiple of τ (G). Therefore, g((u, v)) g((w, z)) = 0. Case : Assume that d G H ((u, v), (w, z)) =. Necessarily, d G (u, w) and d H (v, z). Thus, f 1 (u) f 1 (w) 1 so there exist a f 1 (u) and b f 1 (w) such that a b. Furthermore, since d H (v, z), we may assume there exist 0 i < j k 1 such that f (v) = iτ (G) + 1 and f (z) = jτ (G) + 1. We have already seen that this implies a + iτ (G) + 1 b + jτ (G) + 1 since i < j and 1 a, b τ (G). Therefore, g((u, v)) g((w, z)) 1. Note that for P 3 P 3, we can find a -tone 6-coloring as shown in Figure 10. This coloring is best possible since P 3 P 3 contains C 4 and τ (C 4 ) = 6. However, in this case Theorem 5 gives bounds of 5 = maxτ (P 3 P 3 ), τ (P 3 P 3 ) τ (P 3 P 3 ) minτ (G)χ(H ), χ(g )τ (H) = 15. This alone shows that perhaps a better upper bound exists in terms of other graph properties. On the other hand, K 3 K 3 K 9 so we know τ (K 3 K 3 ) = 18 as seen in Figure 11. In this particular case, we have 6 = minτ (K 3 K 3 ), τ (K 3 K 3 ) τ (K 3 K 3 ) minτ (G)χ(H ), χ(g )τ (H) = 18, which shows the upper bound in Theorem 5 is sharp. References [1] [] A. Bickle and B. Phillips. t-tone Colorings of Graphs (Submitted) [3] D. Cranston, J. Kim, and W. Kinnersley. New Results in t-tone colorings of graphs. preprint, 011. [4] J. Fonger, B. Phillips, and C. Segroves. Math 6450 Final Report

19 1,4,3 5,6 3,5 1,6,4 1, 3,4 1,5 Figure 10: P 3 P 3 13,14 15,16 17,18 7,8 9,10 11,1 1, 3,4 5,6 Figure 11: K 3 K 3 19