Improved Bounds on the L(2, 1)-Number of Direct and Strong Products of Graphs

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1 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR AERS VOL. ***, NO. ***, *** 2007 Improed Bounds on the L(2, )-Number of Direct and Strong roducts of Graphs Zhendong Shao, Sandi Klažar, Daid Zhang, Senior Member IEEE Abstract The frequency assignment problem is to assign a frequency which is a nonnegatie integer to each radio transmitter so that interfering transmitters are assigned frequencies whose separation is not in a set of disallowed separations. This frequency assignment problem can be modelled with ertex labelings of graphs. An L(2, )-labeling of a graph G is a function f from the ertex set V (G) to the set of all nonnegatie integers such that f(x) f(y) 2 if d(x, y) = and f(x) f(y) if d(x, y) = 2, where d(x, y) denotes the distance between x and y in G. The L(2, )-labeling number λ(g) of G is the smallest number k such that G has an L(2, )-labeling with max{f() : V (G)} = k. This paper considers the graph formed by the direct product and the strong product of two graphs and gets better bounds than those of [4] with refined approaches. Index Terms channel assignment, L(2, )-labeling, graph direct product, graph strong product I. INTRODUCTION THE frequency assignment problem is to assign a frequency which is a nonnegatie integer to each radio transmitter so that interfering transmitters are assigned frequencies whose separation is not in a set of disallowed separations. Hale [0] formulated this into a graph ertex coloring problem. In a priate communication with Griggs, Roberts proposed a ariation of the channel assignment problem in which close transmitters must receie different channels and ery close transmitters must receie channels that are at least two channels apart. To translate the problem into the language of graph theory, the transmitters are represented by the ertices of a graph; two ertices are ery close if they are adjacent and close if they are of distance 2 in the graph. Motiated by this problem, Griggs and Yeh [9] proposed the following labeling on a simple graph. An L(2, )-labeling of a graph G is a function f from the ertex set V (G) to the set of all nonnegatie integers such that f(x) f(y) 2 if d(x, y) = and f(x) f(y) if d(x, y) = 2, where d(x, y) denotes the distance between x and y in G. A k-l(2, )-labeling is an L(2, )-labeling such that no label is greater than k. The L(2, )-labeling number of G, denoted by λ(g), is the smallest number k such that G has a k-l(2, )-labeling. Manuscript receied February ***, Zhendong Shao is with the Department of Computer Science and Technology, Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen, 58055,. R. China ( zhdshao0026@6.com. Corresponding author.) Sandi Kla zar is with the Department of Mathematics and Computer Science, Uniersity of Maribor, Koroška cesta 60, 2000 Maribor, Sloenia, and also with the Institute of Mathematics, hysics and Mechanics, Jadranska 9, 000 Ljubljana, Sloenia ( sandi.klazar@uni-mb.si). Supported in part by the Ministry of Science of Sloenia under the grant Daid Zhang is with the Department of Computing, Hong Kong olytechnic Uniersity, Kowloon, Hong Kong. From then on, a large number of articles hae been published deoted to the study of the frequency assignment problem and its connections to graph labelings, in particular, to the class of L(2, )-labelings and its generalizations: Oer 00 references on the subject are proided in a ery comprehensie surey []. In addition to graph theory and combinatorial techniques, other interesting approaches in studying these labelings include: neural networks [7], [5], genetic algorithms [8], and simulated annealing [5], [9]. Most of these papers are considering the alues of λ on particular classes of graphs. From the algorithmic point of iew it is not surprising that it is N-complete to decide whether a gien graph G allows an L(2, )-labeling of span at most n [9]. Hence good lower and upper bounds for λ are clearly welcome. For instance, if G is a diameter 2 graph, then λ(g) 2. The upper bound is attainable by Moore graphs (diameter 2 graph with order 2 + ), see [9]. (Such graphs exist only if = 2,, 7, and possibly 57.) The aboe considerations in particular motiated Griggs and Yeh [9] to conjecture that for any graph G with the maximum degree 2 the best upper bound on λ(g) is 2. (Note that this is not true for =. For example, (K 2 ) = but λ(k 2 ) = 2.) They proided an upper bound for general graphs with maximum degree. Chang and Kuo [4] improed the bound to 2 + while Král and Skrekoski [7] further reduced the bound to 2 +. Furthermore, in 2005, Gonçales [8] announced the bound At the present moment, a journal paper containing a proof of the claimed bound is still to be published. Graph products play an important role in connecting arious useful networks and they also sere as natural tools for different concepts in many areas of research. To justify the first assertion we mention that the diagonal mesh with respect to multiprocessor network is representable as the direct product of two odd cycles [22] while for the other assertion we recall that one of the central concepts of information theory, the Shannon capacity, is most naturally expressed with the strong product of graphs, cf. [2]. In [4], upper bounds and some explicit labelings for the direct product and the strong of graphs and proed which in particular implies that the L(2,)-labeling number of the product graph is bounded by the square of its maximum degree. Hence Griggs and Yeh s conjecture holds in both cases (with some minor exception). The main purpose of this paper is to improe the upper bounds obtained in [4]. The main tool for this purpose is a more refined analysis of neighboorhoods in product graphs than the analysis in [4]. In the next section a heuristic labeling algorithm is presented

2 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR AERS VOL. ***, NO. ***, *** that forms the basis for these considerations while in Sections and 4 direct products and strong products of graphs are considered, respectiely. IMproements with respect to the preiously known upper bounds are explicitly computed. II. A LABELING ALGORITHM A subset X of V (G) is called an i-stable set (or i- independent set) if the distance between any two ertices in X is greater than i. An -stable (independent) set is a usual independent set. A maximal 2-stable subset X of a set Y is a 2-stable subset of Y such that X is not a proper subset of any 2-stable subset of Y. Chang and Kuo [4] proposed the following algorithm to obtain an L(2,)-labeling and the maximum alue of that labeling on a gien graph. Algorithm 2.. Input: A graph G = (V, E). Output: The alue k is the maximum label. Idea: In each step i, find a maximal 2-stable set from the unlabeled ertices that are distance at least two away from those ertices labeled in the preious step. label all the ertices in that 2-stable with i in current stage. The label i starts from 0 and then increases by in each step. The maximum label k is the final alue of i. Initialization: Set X = ; V = V (G); i = 0. Iteration: ) Determine Y i and X i. Y i = {x V : x is unlabeled and d(x, y) 2 for all y X i }. X i a maximal 2-stable subset of Y i. If Y i = then set X i =. 2) Label the ertices in X i (if there is any) by i. ) V V X i. 4) If V, then i i +, go to Step. 5) Record the current i as k (which is the maximum label). Stop. Thus k is an upper bound on λ(g). We would like to find a bound in terms of the maximum degree (G) of G analogous to the bound in terms of the chromatic number χ(g). Let x be a ertex with the largest label k obtained by Algorithm 2.. Set I = {i : 0 i k and d(x, y) = for some y X i }, I 2 = {i : 0 i k and d(x, y) 2 for some y X i }, and I = {i : 0 i k and d(x, y) for all y X i }. It is clear that I 2 + I = k. For any i I, x / Y i ; otherwise X i {x} is a 2-stable subset of Y i, which contradicts the choice of X i. That is, d(x, y) = for some ertex y in X i ; i.e., i I. So, I I. Hence k I 2 + I I 2 + I. In order to find k, it suffices to estimate B = I + I 2 in terms of (G). We will inestigate the alue B with respect to a particular graph. The notations which hae been introduced in this section will also be used in the following sections. III. THE DIRECT RODUCT OF GRAHS In this section, we obtain an upper bound for the L(2, )- labeling number of the direct product of two graphs in terms of the maximum degrees of the graphs inoled. The direct product G H of two graphs G and H is the graph with ertex set V (G) V (H), in which the ertex (, w) is adjacent to the ertex (, w ) if and only if is adjacent to and w is adjacent to w. See Figure for an example. C 4 C4 Fig.. Direct product C 4. Suppose G and H are graphs with (G) = 0 or (H) = 0., by the definition of the direct product, G H contains no edges. Therefore we assume in the rest of this section that (G) and (H). Here is the main result of this section. Theorem.: Let,, and 2 be maximum degrees of G H, G, and H, respectiely. λ(g H) 2 + ( + 2 )( )( 2 ). roof: Let x = (u, ) in V (G) V (H). deg G H (x) = deg G (u)deg H (). Denote d = deg G H (x), d = deg G (u), d 2 = deg H (), = (G) and 2 = (H). Hence d = d d 2 and = (G H) = 2. For any ertex u in G with distance 2 from u, there must be a path u u u of length two between u and u in G. REcall that the degree of in H is d 2, i.e., has d 2 adjacent ertices in H. by the definition of a direct product G H, there must be d 2 internally-disjoint paths of length two between (u, ) and (u, ) in G H. Hence for any ertex in G with distance 2 from u, there must be corresponding d 2 ertices in G H with distance 2 from x = (u, ) which are coincided in G H; on the other hand, wheneer there is not such a ertex in G with distance 2 from u in G, there will neer exist such corresponding d 2 ertices with distance 2 from x = (u, ) which are coincided in G H. In the former case, since such d 2 ertices with distance 2 from x = (u, ) are coincided in G H and hence they can only be counted once, we hae to subtract d 2 from the alue d( ) ( the number d( ) is best possible); in the latter case, since there do not exist such d 2 ertices with distance 2 from x = (u, ) which are coincided in G H at all and hence they must be counted zero, we hae to subtract d 2 from the alue d( ). Let the number of ertices in G with distance 2 from u be t, then t [0, d ( )]. The minimum number we hae to subtract from the alue d( ) in this sense occurs when

3 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR AERS VOL. ***, NO. ***, *** 2007 t = d ( ) and we can get that in this sense the number of ertices with distance 2 from x = (u, ) in G H will decrease at least d ( )(d 2 ) from the alue d( ). (We should notice that the bound d( ) includes the case d ( )d 2.) See Figure 2 for the illustration of the aboe argument. In the figure d() denotes the degree of in H, that is, d() = d 2. u u'' u' Define f(s, t) = st( 2 + ) s( )(t ) t( 2 )(s ). f(s, t) has the absolute maximum at (, 2 ) on [0, ] [0, 2 ] and its alue is f(, 2 ) = 2 + ( + 2 )( )( 2 ). 2 ( u, ) ( u', ) ( u'', ) λ(g H) k B 2 + ( + 2 )( )( 2 ) and the proof is complete. Corollary.2: Let be the maximum degree of G H. λ(g H) 2 except if one of (G) and (H) is. roof: Suppose 2 and 2 2. This implies ( + 2 )( )( 2 ) 2 = ( + 2 )(( )( 2 ) ) 0. ( u'', ) d( ) d( ) Fig. 2. Situation from the proof of Theorem.. By the commutatiity of the direct product we also infer that the number of ertices of distance 2 from x = (u, ) in G H will still decrease d 2 ( 2 )(d ) from the alue d( ). Hence the number of ertices with distance 2 from x = (u, ) in G H will decrease d ( )(d 2 )+d 2 ( 2 )(d ) from the alue d( ) altogether. Hence for the ertex x, the number of ertices with distance from x is not greater than, and the number of ertices with distance 2 from x is not greater than d( ) d ( )(d 2 ) d 2 ( 2 )(d ) = d d 2 ( 2 ) d ( )(d 2 ) d 2 ( 2 )(d ). Hence I d and I 2 d + d( ) d ( )(d 2 ) d 2 ( 2 )(d ) = d d ( )(d 2 ) B = I + I 2 d 2 ( 2 )(d ). d + d d ( )(d 2 ) d 2 ( 2 )(d ) = d( + ) d ( )(d 2 ) d 2 ( 2 )(d ) = d d 2 ( 2 + ) d ( )(d 2 ) d 2 ( 2 )(d ). λ(g H) k B ( + 2 )( )( 2 ) 2. Therefore the result follows. Note that when and 2 are sufficiently large, 2 is a good aproximation for ( )( 2 ). Hence in such cases a good aproximation for the bound of Theorem. is gien by 2 + ( + 2 ) = 2 ( + 2 ). In [4] it is proed that λ(g H) 2 max{( ) 2 ( 2 ) () 2 +, ( )( 2 ) }. We conclude the section by demonstrating that the upper bound of Theorem. is an improement of (). Because 2 max{( ) 2 ( 2 ) 2 +, ( )( 2 ) } ( ( + 2 )( )( 2 )) = min{ ( ) 2 ( 2 ) + + 2, ( )( 2 ) } 2 + ( + 2 )( )( 2 ) = min{ ( ) 2 ( 2 ) + + 2, ( )( 2 ) } + ( + 2 )(( )( 2 ) ) = min{ ( ) 2 ( 2 ) + ( + 2 )( )( 2 ), ( )( 2 ) 2 + ( + 2 )( )( 2 )} = min{ 2 ( )( 2 ), ( )( 2 )},

4 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR AERS VOL. ***, NO. ***, *** we hae thus reduced () by min{ 2 ( )( 2 ), ( )( 2 )}. IV. THE STRONG RODUCT OF GRAHS The strong product G H of graphs G and H is the graph with ertex set V (G) V (H), in which the ertex (, w) is adjacent to the ertex (, w ) if and only if = and w is adjacent to w, or w = w and is adjacent to, or is adjacent to and w is adjacent to w. See Figure for an example. hence they can only be counted once, we hae to subtract d 2 + from the alue d( ) ( the number d( ) is the best possible); on the latter case, since there do not exist such d 2 + ertices with distance 2 from x = (u, ) which are coincided in G H at all and hence they must be counted zero, we hae to subtract d 2 + from the alue d( ). Let the number of ertices in G with distance 2 from u be t, then t [0, d ( )]. The minimum number we hae to subtract from the alue d( ) in this sense occurs when t = d ( ) and we can get that in this sense the number of ertices with distance 2 from x = (u, ) in G H will decrease at least d ( )(d 2 + ) = d ( )d 2 from the alue d( ). (We should notice that the bound d( ) includes the case d ( )d 2.) See Figure 4 for an example of the discussion in this paragraph, where d() denotes the degree of in H, i.e., d() = d u u'' u' Fig.. Strong product 4 By the definition of the strong product G H of two graphs G and H, if (G) = 0 or (H) = 0, then G H consists of disjoint copies of H or G. Thus λ(g H) = λ(h) or λ(g H) = λ(g). Therefore we assume (G) and (H). In [] and [6] the λ-numbers of the strong product of cycles are considered. In this section, we obtain a general upper bound for the λ-number of strong products in terms of maximum degrees of the factor graphs (and the product). Theorem 4.: Let,, and 2 be the maximum degree of G H, G, and H, respectiely. λ(g H) 2 + ( ) 2. roof: Let x = (u, ) in V (G) V (H). deg G H (x) = deg G (u) + deg H () + deg G (u)deg H (). Denote d = deg G H (x), d = deg G (u), d 2 = deg H (), = (G) and 2 = (H). Hence d = d + d 2 + d d 2 and = (G H) = For any ertex u in G with distance 2 from u, there must be a path u u u of length two between u and u in G; but the degree of in H is d 2, i.e., has d 2 adjacent ertices in H, by the definition of a strong product G H, there must be d 2 + internally-disjoint paths of length two between (u, ) and (u, ). Hence for any ertex in G with distance 2 from u, there must be corresponding d 2 + ertices with distance 2 from x = (u, ) which are coincided in G H; on the contrary wheneer there is not such a ertex in G with distance 2 from u in G, there will neer exist such corresponding d 2 + ertices with distance 2 from x = (u, ) which are coincided in G H. In the former case, since such d 2 + ertices with distance 2 from x = (u, ) are coincided in G H and 2 ( u, ) ( u'', ) ( u', ) ( u'', ) d( ) d( ) Fig. 4. Situation from the proof of Theorem 4. ( u'', ) For H, we can analyze similarly and get that the number of ertices with distance 2 from x = (u, ) in G H will still decrease d 2 ( 2 )(d + ) = d 2 ( 2 )d from the alue d( ). Hence the number of ertices with distance 2 from x = (u, ) in G H will decrease d ( )d 2 + d 2 ( 2 )d = ( + 2 2)d d 2 from the alue d( ) altogether. By the aboe analysis, the number d( ) ( + 2 2)d d 2 is now the best possible for the number of ertices with distance 2 from x = (u, ) in G H. Moreoer, by the definition of the strong product we can again analyze as follows: Denote ε, the number of edges of the subgraph F induced by the neighbors of x. The edges of the subgraph F induced by the neighbors of x can be diided into the following three cases. Case : Edges between (u, ) and (u, ), where (u, u) E(G) and (, ) E(H). There are totally d neighbors (u, ) (where u is adjacent to u in G) of x = (u, ) and totally d 2 neighbors (u, ) (where is adjacent to in H) of x = (u, ). Hence the number of edges of the subgraph F

5 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR AERS VOL. ***, NO. ***, *** induced by the neighbors of x is at least d d 2. See Figure 5 for an example. u u' u u' ( u, ) ( u', ) ( u, ) ( u', ) ' ( u, ' ) ( u', ' ) Fig. 7. Situation from the third case Fig. 5. ' ( u, ' ) Situation from the first case Case 2: Edges between (u, ) and (u, ), where (u, u) E(G) and (, ) E(H). There are totally d d 2 neighbors (u, ) (where u is adjacent to u in G and is adjacent to in H) of x = (u, ). Hence the number of edges of the subgraph F induced by the neighbors of x should again add least d d 2 apart from the edges in case. See Figure 6 for an example. u u' distance 2 from x is not greater than Hence I d and d( ) ( + 2 2)d d 2 6d d 2 = (d + d 2 + d d 2 )( ) ( )d d 2. I 2 d + d( ) ( )d d 2 = d ( )d d 2. B = I + I 2 ( u, ) ( u', ) d + d ( )d d 2 = d( + ) ( )d d 2 = (d + d 2 + d d 2 )( ) ( )d d 2. Define Fig. 6. ' Situation from the second case ( u, ' ) ( u', ' ) Case : Edges between (u, ) and (u, ), where (u, u) E(G) and (, ) E(H). There are totally d d 2 neighbors (u, ) (where u is adjacent to u in G and is adjacent to in H) of x = (u, ). Hence the number of edges of the subgraph F induced by the neighbors of x should again add least d d 2 apart from the sum of the edges in case and case 2. See Figure 7 for an example. The present upper bound on the number of ertices at distance two from x is d( ) ( + 2 2)d d 2. For each edge in F, this upper bound is decreased by 2. Hence, by the analysis of the aboe three cases, the number of ertices with distance 2 from x = (u, ) in G H will still need to decrease by at least 6d d 2. (the number d( ) ( + 2 2)d d 2 is now the best possible for the number of ertices with distance 2 from x = (u, ) in G H.) Hence for the ertex x, the number of ertices with distance from x is not greater than. The number of ertices with f(s, t) = (s + t + st)( ) ( )st. On [0, ] [0, 2 ] the function f(s, t) has the absolute maximum at (, 2 ) and the alue f(, 2 ) is equal ( )( ) ( ) 2 = ( + ) ( ) 2 = 2 + ( ) 2 = 2 + ( ) ( ) 2. λ(g H) k B 2 + ( ) 2 and we are done. Corollary 4.2: Let be the maximum degree of G H. λ(g H) 2 if both (G) and (H) are, otherwise λ(g H) 2 9. roof: Case. If both and 2 are, then the connected components of G H are K, K 2, and K 4, hence λ(g H) = λ(k 4 ) = 6 = 9 = 2.

6 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR AERS VOL. ***, NO. ***, *** Case 2. Suppose at least one of and 2 is greater than. This implies ( )( 2 ) = ( ) 2 ( ) + 6 = 6. ( ) ( ) 2 5. Hence λ(g H) k B 2 + ( ) ( ) 2 = 2 + ( ) ( ) [].K. Jha, Optimal L(2,)-labeling of strong product cycles, IEEE Trans. Circuits & Systems I: Fundamental Theory and Appl. 48, (200), [4] S. Kla zar and S. Spacapan, The 2 -conjecture for L(2,)-labelings is true for direct and strong products of graphs, IEEE Trans. Circuits and Systems II: Express Briefs 5, (2006), [5] N. Kunz, Channel assignment for cellular radio using neural networks, IEEE Trans. On Vehicular Technology, 40, (992), [6] D. Kor ze and A. Vesel, L(2, )-labeling of strong products of cycles, Information rocessing Letters, 94, (2005), [7] D. Král and R. Skrekoski, A theorem about channel assignment problem, SIAM J. Discrete Math. 6, (200), [8] W.K. Lai and G.G. Coghill, Channel assignment through eolutionary optimization, IEEE Trans. On Vehicular Technology, 45, (996), [9] R. Mathar, Channel assignment in cellular radio networks, IEEE Trans. On Vehicular Technology, 42, (99), [20] F. S. Roberts, T-colorings of graphs: Recent results and open problems, Discrete Math., 9, (99), [2] Z. Shao and R. K. Yeh, The L(2, )-labeling and operations of graphs, IEEE Trans. Circuits & Systems I: Regular apers, 52, (2005), [22] K. W. Tang and S. A. adubirdi, Diagonal and toroidal mesh networks, IEEE Trans. Comput., 4, (994), [2] A. Vesel and J. Žeronik, Improed lower bound on the Shannon capacity of C 7, Inform. roc. Lett., 8, (2002), and the proof is complete. In [4] it is proed that Because λ(g H) ( ( ) 2 ) = ( ) 2 = ( + 2 2) 2, we reduce the bound by ( + 2 2) 2. REFERENCES [] S. S. Adams, J. Cass and D. S. Troxell, An Extension of the Channel-Assignment roblem: L(2, )-Labelings of Generalized etersen Graphs, IEEE Trans. Circuits & Systems I: Regular apers, 5 (2006), [2] H. L. Bodlaender, T. Kloks, R. B. Tan and J.. Leeuwen, λ-coloring of graphs, Lecture Notes in Computer Science, 770, Springer-Verlag Berlin, Heidelberg, (2000), [] T. Calamoneri, The L(h, k)-labelling roblem: A Surey and Annotated Bibliography, The Computer Journal, 49, (2006) [4] G. J. Chang and D. Kuo, The L(2, )-labeling on graphs, SIAM J. Discrete Math. 9, (996), [5] M. Duque-Anton, N. Kunz and B. Ruber, Channel assignment for cellular radio using simulated annealing, IEEE Trans. On Vehicular Technology, 42, (99), 4 2. [6] J. Fiala, T. Kloks and J. Kratochíl, Fixed-parameter complex of λ- labelings, Lecture Notes in Computer Science, 665, Springer-Verlag Berlin, Heidelberg, (999), [7] N. Funabiki and Y. Takefuji, A neural network parallel algorithm for channel assignment problem in cellular radio networks, IEEE Trans. On Vehicular Technology, 4, (992), [8] D. Gonçales, On the L(p, )-labelling of graphs, in 2005 European Conference on Combinatorics, Graph Theory and Applications (Euro- Comb 05), Stefan Felsner (ed.), Discrete Mathematics and Theoretical Computer Science roceedings AE, pp [9] J. R. Griggs and R. K. Yeh, Labeling graphs with a condition at distance two, SIAM J. Discrete Math. 5, (992), [0] W. K. Hale, Frequency assignment: Theory and application, roc. IEEE, 68, (980), [] W. Imrich and S. Klažar, roduct Graphs: Structure and Recognition, Wiley, New York, [2].K. Jha, Optimal L(2,)-labelings of Cartesian products of cycles with an application to independent domination, IEEE Trans. Circuits & Systems I: Fundamental Theory and Appl. 47, (2000), 5 54.