Packing Chromatic Number of Cycle Related Graphs

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1 International Journal of Mathematics and Soft Computing Vol., No. (0), 7 -. ISSN Print : 9-8 ISSN Online: 9 - Packing Chromatic Number of Cycle Related Graphs Albert William, S. Roy Department of Mathematics, Loyola College, Chennai 600 0, INDIA. sroysantiago@gmail.com Indra Rajasingh School of Advanced Sciences, VIT University, Chennai 600 7, INDIA. Abstract The packing chromatic number χ ρ (G) of a graph G is the smallest integer k for which there exists a mapping π : V (G) {,,..., k} such that any two vertices of color i are at distance at least i +. It is a frequency assignment problem used in wireless networks, which is also called Broadcast coloring. It is proved that packing coloring is NP-complete for general graphs and even for trees. In this paper, we give the packing chromatic number for some classes of cycles. Keywords: Packing chromatic number, kc n - linear, barbell graph, split graph, total graph, crown graph, lollipop, tadpole. AMS Subject Classification(00): 0C. Introduction Let G be a connected graph and k be an integer, k. A packing k-coloring of a graph G is a mapping π : V (G) {,,..., k} such that any two vertices of color i are at distance at least i +. Thus, the vertices of G are partitioned into different color classes X, X,.., X k, where every X i is an i-packing of G. The i-packing number of G, denoted by ρ i (G), is the maximum cardinality of an i-packing that occurs in G. The packing chromatic number χ ρ (G) of G is the smallest integer k for which G has packing k-coloring. The concept of packing coloring emerges from the area of frequency assignment in wireless networks and was introduced by Goddard et al. [8] by the name Broadcast coloring. It has several applications such as resource replacement and biological diversity. The term packing chromatic number was introduced by Bresar []. Goddard et al. [8] proved that the packing coloring problem is NP-complete for general graphs and Fiala and Golovach [6] proved that it is NP-complete even for trees. It is proved that packing coloring problem is solvable in polynomial time for graphs whose treewidth and diameter are both bounded [6] and for cographs and split graphs [8]. Sloper [] studied a special type of packing coloring, called eccentric chromatic coloring and proved that the infinite -regular tree has packing chromatic number 7. For the infinite planar square lattice Z, 0 χ ρ (Z ) 7 [, 9]. The packing coloring of distance graphs were studied in [, ]. For the infinite hexagonal lattice H, χ ρ (H) = 7 []. Argiroffo et al. [, ] proved that the packing coloring is solvable in polynomial time for the class of (q, q ) graphs, partner limited graphs and for an infinite subclass of lobsters, 7

2 8 Albert William, Indra Rajasingh and S. Roy including caterpillars. In [7, ] it is proved that the infinite, planar triangular lattice and the three dimensional square lattice have unbounded packing chromatic number. In this paper, we study the packing chromatic number of kc n - linear, barbell graph, split graph, total graph, crown graph, lollipop and tadpole. Main Results Lemma.. [8] Let H be a subgraph of G. Then χ ρ (H) χ ρ (G). { when n is a multiple of Lemma.. [8] χ ρ (C n ) = when n is not a multiple of. Lemma.. [8] Let G be a complete graph on n vertices. Then χ ρ (G) = n. Definition.. A graph is called a kc n - linear if it is a connected graph with k blocks, each of which is isomorphic to C n. Moreover, if u and v are cut-vertices in the same block B i, then d(u, v) = n. Theorem.. For kc n, n, χ ρ (kc n ) = when n 0 mod. Proof: Case : n = 8r. We begin with the cut-vertex common to B i, B i+ and label it as. Label every -sequence of consecutive vertices on B i in the clockwise sense as,,, beginning from the cut-vertex. See Figure (a). Figure : (a) χ ρ (C 8 ) = (b) χ ρ (C ) =. Case : n = 8r +. Label the cut-vertex common to B i, B i+ as or according as i is odd or even respectively. Label every -sequence of consecutive vertices on B i in the clockwise sense as,,, or,,, beginning from the cut-vertex with label or. See Figure (b). By Lemma. and., χ ρ (kc n ) =. Definition.6. [] A gear graph is obtained from the wheel W n by adding a vertex between every pair of adjacent vertices of the n-cycle and it is denoted by G n, where n is the number of vertices of wheel graph. G n has n + vertices and n edges. Theorem.7. For the gear graph G n, χ ρ (G n ) =, when n is even. Proof: Let u and v be the vertices of G n such that deg(u) = n and deg(v) =. Color the vertices in the outercycle with the sequence, starting from the vertex v in any direction and fix color to the vertex u. Thus, χ ρ (G n ).

3 Packing Chromatic Number of Cycle Related Graphs 9 Figure : χ ρ (G 6 ) = and χ ρ (G 8 ) =. By Lemma. and., χ ρ (G n ) =. Definition.8. [6] For a graph G, the splitting graph S of G is obtained by adding a new vertex v corresponding to each vertex v of G such that N(v) = N(v ). Theorem.9. Let K,n, n be a star on n vertices. Then χ ρ (S (K,n )) =. Proof: Let v, v,..., v n be the pendent vertices and v be the apex vertex of K,n and u, u, u,..., u n be the added vertices corresponding to v, v, v,..., v n to obtain S (K,n ). u u u u u u6 u7 v v v v v v v6 v7 u Figure : χ ρ (S (K,7 )) =. Color the vertices u, u,..., u n and v, v,..., v n with color. Assign color to the vertex u and to the vertex v. Thus χ ρ (S (K,n )). By Lemma. and., χ ρ (S (K,n )) =. Definition.0. [] The n-barbell graph is a simple graph obtained by connecting two copies of a complete graph K n by a bridge and it is denoted by B(K n, K n ). Theorem.. For B(K n, K n ), χ ρ (B(K n, K n )) = n. Proof: An n-barbell graph contains two copies of K n namely K n and K n. Let uv be the connecting bridge where u K n and v K n. We give an algorithm to color B(K n, K n ) using exactly n colors. Procedure PACKING COLORING B(K n, K n ) Input: Barbell graph B(K n, K n ) Algorithm:

4 0 Albert William, Indra Rajasingh and S. Roy Step : Label the vertex u with color n. Step : Label the vertex v with color n. Step : Label any vertex of K n and K n with color. Step : Label any vertex of K n and K n with color. Step : Label the remaining vertices of K n and K n with color,,, n. Output: χ ρ (B(K n, K n ))) = n. 6 Figure : χ ρ (B(K, K ))) = 6. Proof of correctness: Let G B(K n, K n ). The diameter of G is, which implies ρ (G) = and ρ i (G) for all integer i. Case : If vertex u or v is labeled by color, then n vertices at distance greater than one from the vertex u or v form a subgraph, which is isomorphic to K n. In this case, at most one vertex receives color, because ρ (K n ) =. Case : If a vertex of G except u and v is labeled by color, then n vertices at distance more than one from that vertex received color form a subgraph, which is isomorphic to K n. In this case, at most one vertex receives color, because ρ (K n ) =. So, ρ (G) =. There is a k-coloring of the graph G with X = and X =. As for each of the other (n ) vertices, one new color is necessary, which implies that χ ρ (B(K n, K n )) = n. Definition.. The graph obtained by joining a pendent edge at each vertex of a cycle C n is called a crown graph denoted by C n K. Theorem.. For C n K, n 6. χ ρ (C n K ), when n 0, (mod ). Proof: Let the vertices of C n be u, u,..., u n and the vertices of pendent edges be v, v,..., v n where u i and v j are adjacent for all i = j. Case : n 0 mod. Color the vertices u, u,..., u n with the sequence, starting from vertex u in the clockwise sense and v, v,..., v n with the sequence, starting from vertex v in the clockwise sense. Thus χ ρ (C n K ). Case : n mod. Color the vertices u, u,..., u n with the sequence, starting from vertex u in the clockwise sense and v, v,..., v n with the sequence, starting from vertex v in the clockwise sense. Thus χ ρ (C n K ).

5 Packing Chromatic Number of Cycle Related Graphs Figure : χ ρ (C 8 K ) = and χ ρ (C 8 K ) =. Definition.. The tadpole graph T (m, n) is obtained from a cycle C m and a path P n, by joining one of the end vertices of P n to a vertex of C m. { Theorem.. For T (m, n), n, χ ρ (T (m, n)) = Proof: Case : m 0 mod., when m 0 mod, when m 0 mod. Let u be a vertex in T (m, n) with deg(u) =. Color the vertices of cycle C m with the sequence, starting from the vertex u in the clockwise sense. Let v be a vertex of path P n which is adjacent to vertex u. Color the path P n with the sequence, starting from the vertex v. Thus χ ρ (T (m, n)). By Lemma. and., χ ρ (T (m, n)) =. Figure 6: χ ρ (T(8,)) =,χ ρ (T(6,)) =,χ ρ (T(7,)) = and χ ρ (T(9,)) =. Case : m 0 mod Let u be a vertex in T (m, n) with deg(u) =. Color the vertices of cycle C m with color sequence in the clockwise sense starting from vertex u with repeated blocks of,,, with an adjustment at the very end as shown below:,,..., when n = r +,,..., when n = r +,,..., when n = r + starting from the vertex u in the clockwise sense. Let v be the vertex of path P n which is adjacent to vertex u. Color the path

6 Albert William, Indra Rajasingh and S. Roy P n with the sequence, starting from the vertex v. Thus χ ρ (T (m, n)). By Lemmas. and., χ ρ (T (m, n)) =. Definition.6. [0] The lollipop graph LP (n, k) is obtained from a complete graph K k and a path P n k+, by joining one of the end vertices of P n k+ to a vertex of K k. Theorem.7. For lollipop graph LP (n, k), χ ρ (LP (n, k)) = n. Proof: Let u be the vertex in LP (n, k) with deg(u) = n. Give color n to the vertex u and colors to n to the remaining vertices in K k. 6 Figure 7: χ ρ (LP (6, )) = 6. Let v be the vertex of path P n k+ which is adjacent to the vertex u. Color the path P n k+ with the sequence, starting from the vertex v. Thus χ ρ (LP (n, k)) n. By Lemma. and., χ ρ (LP (n, k)) = n. Acknowledgment This work is supported by Maulana Azad Fellowship F-7./0/MANF-CHR-TAM- of the University Grants Commission, New Delhi, India. References [] G. Argiroffo, G. Nasini and P. Torres, The Packing Coloring of Lobsters and Partner Limited Graphs, Discrete Applied Mathematics. Preprint (0). [] G. Argiroffo, G. Nasini and P. Torres, The Packing Coloring Problem for (q, q ) Graphs, Combinatorial Optimization, Lecture Notes in Computer Science 7(0), [] B. Brešar, S. Klavžar and D. F. Rall, On the Packing Chromatic Number of Cartesian Products, Hexagonal Lattice and Trees, Discrete Appl. Math, (007), 0-. [] J. Ekstein, P. Holub and B. Lidický, Packing Chromatic Number of Distance Graphs, Discrete Applied Mathematics, 60 (0), 8-. [] J. Fiala, S. Klavžar, B. Lidický, The Packing Chromatic Number of Infinite Product G, European J. Combin., 0 () (009), 0-. [6] J. Fiala and P. A. Golovach, Complexity of the packing chromatic problem for trees, Discrete Appl. Math., 8 (00),

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