LOCAL CONNECTIVE CHROMATIC NUMBER OF CARTESIAN PRODUCT OF SOME GRAPHS

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1 LOCAL CONNECTIVE CHROMATIC NUMBER OF CARTESIAN PRODUCT OF SOME GRAPHS ROMANIAN JOURNAL OF MATHEMATICS AND COMPUTER SCIENCE, 08, VOLUME 8, ISSUE, p-7 CANAN C IFTC I AND PINAR DU NDAR Abstract A local connective k-coloring of a graph G is a proper vertex coloring, which assigns colors from {,,, k} to vertices of V (G) in a such way that any two non adjacent vertices u and v of a color i satisfies κ(u, v) > i, where κ(u, v) is the maximum number of internally disjoint paths between u and v Adjacent vertices are colored with different colors as in the proper coloring The smallest integer k for which there exists a local connective k coloring of G is called the local connective chromatic number of G and denoted by χlc (G) In this paper, we determine the local connective chromatic number of Cartesian product of some graphs Mathematics Subject Classification (00): 05C5, 05C76 Key words: Graph coloring, packing chromatic number, internally disjoint path, local connective chromatic number, Cartesian product Article history: Received 0 March 07 Accepted April 08 Introduction Let G = (V (G), E(G)) be a simple undirected graph, where V (G) and E(G) denote the set of vertices and edges of G, respectively Two vertices u and v are adjacent if they are joined by an edge e = uv The degree of v is the number of adjacent vertices of v For the notations and terminology we follow [] A set of paths from a vertex u to a vertex v is said to be internally disjoint (vertex disjoint) if no two paths share a common vertex except u and v The local connectivity κg (u, v) = κ(u, v) between two distinct vertices u and v of a graph G is defined as the smallest number of vertices whose removal separates u and v By Menger s Theorem [], κ(u, v) equals the maximum number of internally disjoint paths between u and v in G It is easy to verify that κ(u, v) 6 min{deg(u), deg(v)} [] Different variants of graph coloring problems have been studied in the literature such as packing coloring, injective coloring, b coloring, dynamic coloring and many more [4, 5, 6, 7, 9] In [], inspiring by the notion of packing coloring [, 4, 8, 4] we define a new coloring concept as local connective coloring and give some bounds on it Further, we study on local connective chromatic number of direct product of paths and cycles [] A local connective k-coloring of a graph G is a mapping c : V (G) {,,, k} such that () If uv E(G), then c(u) 6= c(v), and () If uv / E(G) and c(u) = c(v) = i, then κ(u, v) > i, where κ(u, v) is the maximum number of internally disjoint paths between u and v The smallest integer k for which there exists a local connective k coloring of G is called the local connective chromatic number of G and denoted by χlc (G) The first condition characterizes proper coloring Thus, every local connective coloring is a proper coloring The vertices of G are partitioned into disjoint color classes X, X,, Xk, where each color class Xi

2 n [ consists of distinct vertices u, v Xi such that κ(u, v) > i and Xi = V (G) The maximum cardinality ROMANIAN JOURNAL OF MATHEMATICS AND COMPUTER SCIENCE, 08, VOLUME 8, ISSUE, p-7 i= of Xi in G is denoted by ki By definition of local connective coloring, it is easily said that χlc (Kn ) = n Routing is the process of delivering messages among vertices and selecting the best paths in a network Efficiency and reliability of routing can be achieved by using internally disjoint paths Because the failure of a path would not affect the performance of other paths Hence, the more internally disjoint paths are the better for a network [0] Thus, we color the vertices of the graph depending on the number of internally disjoint paths between two vertices The Cartesian product of two graphs G and H, denoted by GH, has vertex set V (G) V (H) and two vertices (u, v ) and (u, v ) are adjacent if and only if either u = u and v v E(H), or u u E(G) and v = v The graphs GH and HG are isomorphic We consider vertex set of GH as an m n array in which the entry (i, j) corresponds the vertex (ui, vj ) and each column induces a copy of G and each row induces a copy of H, where ui V (G) and vj V (H) In [], C iftc i and Du ndar give the following results for the local connective chromatic number of Pn, Cn and K,n Theorem [] Let Pn be a path of order n with n > Then, χlc (Pn ) = + b n c Theorem [] Let Cn be a cycle of order n with n > Then, (, if n is even χlc (Cn ) =, if n is odd Theorem [] Let K,n, where n >, be a star Then, χlc (K,n ) = Local Connective Chromatic Number of Cartesian Product of Some Graphs In this section, we determine local connective chromatic number of Cartesian product of some known graphs such as path, cycle, star, wheel graph and complete graph Theorem Let Pm and Pn be two paths of order m and n with 6 m 6 n, respectively Then χlc (Pm Pn ) = Proof The graph Pm Pn has four vertices of degree two, (m + n 4) vertices each of degree three and all other vertices of degree four Thus, κ(u, v) 6 4 for any vertices u and v in Pm Pn Also, the graph Pm Pn has n copies of Pm and m copies of Pn By Theorem, χlc (Pn ) = + bn/c Since the number of internally disjoint paths between any two vertices is at most 4, the number of colors needed for coloring Pm in Pm Pn decreases Then, we color the first copy of Pm with two colors as,,,,,, Consider the following coloring pattern Let i and j denote the row and column of a vertex with 6 i 6 m and 6 j 6 n Then assign color to every vertex with i + j even; assign color to every vertex i + j odd (see Fig ) Then, the pattern looks as follows: Consequently, the graph Pm Pn is colored with two colors

3 ROMANIAN JOURNAL OF MATHEMATICS AND COMPUTER SCIENCE, 08, VOLUME 8, ISSUE, p-7 Figure Cartesian product of P4 and P5 Theorem Let Cm and Pn be a cycle and a path of order m and n with 6 m 6 n, respectively Then (, if m is even, χlc (Cm Pn ) =, if m is odd Proof The graph Cm Pn has m vertices of degree and m(n ) vertices each of degree 4 Hence, κ(u, v) 6 4 for all vertices u, v V (Cm Pn ) Further, Cm Pn has n copies of Cm and m copies of Pn By Theorem, it is known that χlc (Pn ) = + bn/c Since the number of internally disjoint paths between the vertices of Pn in Cm Pn increases, the vertices of Pn in Cm Pn are colored with less colors than + bn/c Thus, we color the first copy of Pn with two colors We prove the theorem with two cases depending on n and m being odd or even Case Let m be odd If m is odd, we have χlc (Cm Pn ) > χlc (Cm ) = by Theorem Thus, we start coloring the graph with coloring the first copy of Cm Consider an m n array and fill the first row and column of this array as coloring the first copy of Cm and Pn with three and two colors, respectively Case Let n be odd We color the first and second copy of Cm as,,,,,,, and,,,,,,,, respectively The other copies are colored by repeated blocks of,,,,,,,,,,,,,, Then the pattern

4 ROMANIAN JOURNAL OF MATHEMATICS AND COMPUTER SCIENCE, 08, VOLUME 8, ISSUE, p-7 is a local connective coloring Case Let n be even is a local connective coloring when it is done similarly to Case Case Let m be even It is known that χlc (Cm ) = by Theorem Thus, χlc (Cm Pn ) > χlc (Cm ) = We can start coloring the graph with first copy of Pn or Cm with two colors Consider start coloring the graph with coloring the first copy of Cm with two colors If n is odd or even, the pattern of the first and second copy of Cm is,,,,,, and,,,,,,, respectively The other n copies of Cm are colored by repeated blocks of the first and second copy of Cm, respectively If n is odd, the pattern is a local connective coloring If n is even, the pattern is a local connective coloring Consequently, we have the statement of theorem from Case and Case Theorem Let Cm and Cn be two cycles of order m and n with 6 m 6 n, respectively Then (, if m and n are even χlc (Cm Cn ) = max{χlc (Cm ), χlc (Cn )} =, otherwise Proof The graph Cm Cn is 4 regular graph Then, for all vertices u, v V (Cm Cn ), we have κ(u, v) 6 4 Thus, for all integer i 6 4, it follows that ki (Cm Cn ) > and otherwise ki (Cm Cn ) 6 The graph Cm Cn has n copies of Cm and m copies of Cn There are three cases to consider Case Let m and n be even By Theorem, χlc (Cm ) = χlc (Cn ) = Start coloring the graph with coloring the first copy of Cm with two colors The first and second column of m n array are filled as,,,,,, and,,,,,,, 4

5 ROMANIAN JOURNAL OF MATHEMATICS AND COMPUTER SCIENCE, 08, VOLUME 8, ISSUE, p-7 respectively Since κ(u, v) 6 4 for all vertices u, v in Cm Cn, the other columns are colored with the repeated blocks of the first and second columns Then the pattern is a local connective coloring Case Let m be odd and n be even By Theorem, χlc (Cm ) = and χlc (Cn ) = Thus, we have χlc (Cm Cn ) > Start coloring the graph with coloring the first copy of Cm with three colors and fill the first and second column of m n array as,,,,,,, and,,,,,,, Since n is even and κ(u, v) 6 4 for u, v V (Cm Cn ), the other columns are colored with n repeated blocks of the first and second column Then the pattern is a local connective coloring Case Let m and n be odd Since m is odd and n is even, we color m (n ) subarray of m n array as in Case But, the last column of m n array remains uncolored The vertex (i, n) is adjacent to vertices (i, ) and (i, n ), where i {,,, m} Thus, the entry (i, n), that is the last copy of Cm, is filled as,,,,,,,,, Then the pattern is a local connective coloring Theorem 4 Let K,m and K,n be two stars of order m + and n + with 6 m 6 n Then χlc (K,m K,n ) = Proof By Theorem, χlc (K,m ) = χlc (K,n ) = This means that χlc (K,m K,n ) > The graph K,m K,n has one vertex of degree m + n, n vertices each of degree m +, m vertices each of degree n + and all other vertices of degree Let ui V (K,m ) and vj V (K,n ), where 6 i 6 m +, 6 j 6 n + and let u and v be center vertices of K,m and K,n, respectively Vertex u is adjacent to vertex ui for each i with 6 i 6 m + 5

6 ROMANIAN JOURNAL OF MATHEMATICS AND COMPUTER SCIENCE, 08, VOLUME 8, ISSUE, p-7 and vertex v is adjacent to vertex vj for each j with 6 j 6 n + Consider start coloring the first copy of K,m In the graph K,m K,n, vertex (u, v ) is adjacent to vertex (uk, v ) for each k, 6 k 6 m + Thus, let c(u, v ) = and c(uk, v ) = for 6 k 6 m + For the second copy of K,m, since vertices (uk, v ) and (uk, v ) are adjacent for each k, 6 k 6 m +, these pairs of vertices can not be colored with the same colors Hence c(u, v ) = and since the number of internally disjoint paths between these vertices are at least, we have c(uk, v ) = for 6 k 6 m + None of vertices of the second copy of K,m are joined with vertices of the remaining n copies of K,m Further, since the number of internally disjoint paths between these vertices are at most, the vertices in the remaining n copies receive the same color as vertices of the second copy of K,m Then the pattern is a local connective coloring of K,m K,n Theorem 5 Let Km and Kn be complete graphs of order m and n Then χlc (Km Kn ) = max{m, n} Proof It is known that χlc (Km ) = m and χlc (Kn ) = n The graph Km Kn has mn vertices each of degree m + n Hence, κ(u, v) 6 min{deg(u), deg(v)} = m + n for u, v V (Km Kn ) Let 6 m 6 n There are n copies of Km and m copies of Kn in the graph Km Kn Since m 6 n, we start coloring the first copy of Kn with different n colors, say,,, n Since m + n > n, we use circular permutations on {,,, n} for the other copies of Kn Then we have χlc (Km Kn ) = max{m, n} = n Conclusion In this paper, we study on local connective chromatic number of Cartesian product of some graphs This coloring is introduced by us in [] and the vertices of a graph is colored depending on the maximum number of internally disjoint paths between them Since by Menger s Theorem [] local connectivity between two vertices is equal to the maximum number of internally disjoint paths between these vertices, relation between local connective chromatic number and connectivity can be examined in future study References [] B Bres ar, S Klavz ar, and DF Rall, On the packing chromatic number of Cartesian products, hexagonal lattice, and trees, Discrete Appl Math 55 (007), 0- [] C C iftc i and P Du ndar, Some bounds on local connective chromatic number, New Trends in Mathematical Sciences, 5() (07), 04 [] C C iftc i and P Du ndar, Local connective chromatic number of direct product of paths and cycles, Bulletin of the International Mathematical Virtual Institute, 7 (07), [4] W Goddard, S M Hedetniemi, S T Hedetniemi, J M Harris and D F Rall, Broadcast chromatic numbers of graphs, Ars Combinatoria, 86 (008), 50 [5] G Hahn, J Kratochvı l, J S ira n and D Sotteau, On the injective chromatic number of graphs, Discrete Mathematics, 56(-) (00), 79 9 [6] RW Irving and DF Manlove, The b chromatic number of a graph, Discrete Applied Mathematics, 9 (999), 7 4 [7] R Javadi and O Behnaz, On b coloring of cartesian product of graphs, Ars Combinatoria 07 (0),

7 ROMANIAN JOURNAL OF MATHEMATICS AND COMPUTER SCIENCE, 08, VOLUME 8, ISSUE, p-7 [8] M Kles c and S Schro tter, On the packing chromatic number of semiregular polyhedra, Acta Electrotechnica et Informatica, (0), 7 [9] HJ Lai and B Montgomery, Dynamic coloring of graphs, Department of Mathematics, West Virginia University, 00 [0] CN Lai, Optimal construction of all shortest node-disjoint paths in hypercubes with applications, IEEE Transactons on parallel and Distributed Systems (0), 9 4 [] K Menger, Zur allgemeinen Kurventheorie, Fundementa Mathematicae, 0 (97), 96 5 [] L Volkmann, On local connectivity of graphs, Applied Mathematics Letters, (008), 6 66 [] D B West, Introduction to graph theory (Vol ) Upper Saddle River: Prentice hall, 00 [4] A William and S Roy, Packing chromatic number of cycle related graphs, International Journal of Mathematics and Soft Computing, 4 (04), 7 Department of Mathematics, Faculty of Arts and Sciences, Ordu University, Ordu, Turkey address: cananciftci@oduedutr Department of Mathematics, Faculty of Science, Ege University, Izmir, Turkey address: pinardundar@egeedutr 7