210. Strong Rainbow Coloring of Ladder-Like Networks

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Strong Rainbow Coloring of Ladder-Like Networks

Paul D1, G Vidya2 and Indra Rajasingh2 1 Department of Science & Humanities, Sri Sairam Institute of technology, Chennai - 600044. 2 School of Advanced Sciences, VIT University, Chennai - 600127. [email protected]

Abstract: A rainbow edge coloring of a connected graph is a coloring of the edges of the graph, such that every pair of vertices is connected by at least one path in which no two edges are colored the same. The minimum number of colors required to make G strongly rainbow connected is known as strong rainbow connection number and is denoted by src(G). We give algorithms for strong rainbow edge coloring of ladder-like networks using optimum number of colors. AMS Subject Classiftcation: 05C15

Keywords: rainbow coloring; strong rainbow coloring; ladder-like networks;

1. Introduction Let G be a nontrivial connected graph on which an edge-coloring c: E ( G ) {1,..., n }, n N , is defined, where adjacent edges may be colored the same. A path is a rainbow path if no two edges of it are colored the same. An edge-coloring of graph G is rainbow connected if any two vertices are connected by a rainbow path. An edge coloring under which G is rainbow connected is called a rainbow coloring. An edge colored graph G is strongly rainbow connected if for every pair of distinct vertices, there exists at least one shortest rainbow path. Thus, the rainbow connection number of a connected graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make G rainbow connected and the strong rainbow connection number is the minimum number of colors that makes G strongly rainbow connected and is denoted by src(G). Thus rc(G) ≤ src(G) for every connected graph G. The concept of rainbow coloring was introduced by Chartand et al in 2008 [1] . The rainbow connection number is not only a natural combinatorial measure, but it also has applications to the secure transfer of classified information between agencies. The rainbow connections of graphs are new concepts and there has been great interest in these concepts and a lot of results have been published. Precise values of rainbow connection number for many special graphs like complete multipartite graphs, Peterson graph and wheel graph have also been determined. It was shown in Chakraborty et al [2], that computing the rainbow connection number of an arbitrary graph is NP-Hard. The Department of Homeland Security of USA was created in 2003 in response to the weaknesses discovered in the transfer of classified information after the September 11, 2001 terrorist attacks. Ericksen made the following observation: An unanticipated aftermath of those deadly attacks was the realization that law enforcement and intelligence agencies could not

communicate with each other through their regular channels, from radio systems to databases. The technologies utilized were separate entities and prohibited shared access, meaning that there was no way for officers and agents to cross check information between various organizations. While the information needs to be protected since it relates to national security, there must also be procedures that permit access between appropriate parties. This two-fold issue can be addressed by assigning information transfer paths between agencies which may have other agencies as intermediaries while requiring a large enough number of passwords and firewalls that is prohibitive to intruders, yet small enough to manage. Honeycomb-type torus networks are appealing other options to torus networks due to the smaller node degree , which prompts to lower complexity and lower implementation cost. Stojmenovic proposed three classes of honeycomb-type tori: honeycomb tori, honeycomb rectangular tori, and honeycomb rhombic tori. For honeycomb tori, Stojmenovic tended to their topological properties and exhibited the related routing and broadcasting algorithms. Megson et al. [4] showed that they contain Hamiltonian cycles, even within the sight of node failures, and Yang and Huang demonstrated that they are rotational Cayley graphs. For honeycomb rectangular tori, Parhami and Kwai found that they can be viewed as a pruned version of tori, and they are Cayley graphs. As for honeycomb rhombic tori, Yang et al. shown that they are additionally Cayley graphs. Recently, these honeycomb tori have been perceived as an attractive alternative to existing torus interconnection networks in parallel and distributed applications [4].

2. Circular Ladder

Lemma 2.1. [1] Let G be a nontrivial connected graph of size m. Then 1. src(G) = 1 if and only if G is a complete graph; 2. rc(G) = src(G) = m if and only if G is a tree; n 3. rc(Cn) = src(Cn) = for each integer n ≥ 4, where Cn is a cycle with size n. 2

Definition 2.2. [5] Cartesian product G H of two graphs G and H is the graph with vertex set VGVH()() , two vertices (uu , ' ) and (vv , ') being adjacent if and only if either u = v and uvEH''() or uv'' and uv E() H .

Definition 2.3. [3] The circular ladder CLn of length n = 3 is the Cartesian product CLn = Cn K2.

n Theorem 2.4. Let G be a circular ladder CLn, n ≥ 4. Then src(G) =  n . 2

Proof. Let the vertices in the outer cycle of circular ladder be v1, v2, . . . , vm and the vertices of inner cycle be u1, u2, . . . , um. The vertices of inner cycle and outer cycle are joined by edges called spokes. By Lemma 2.1, If Cn is a cycle on n vertices, n ≥ 4, then src(Cn) = n . Consider the vertices u and v in G. Let us assume that one vertex u is in the outer cycle and the other

vertex v is in the inner cycle. The (u, v)- geodesic passes through exact one spoke and it has to be colored with a distinct color. This implies that every spoke receives a distinct color. Therefore n src(G) =  n . See Figure 1. 2

Figure 1: Strong rainbow coloring of CL11

3. Extended Cycle-of-Ladder

In 2008, Jywe-Fei Fang introduced a network called cycle-of-ladder and proved that it is a spanning subgraph of the hypercube network, thereby proving that hypercube network is bipancyclic [6].

Definition 3.1. The n-ladder graph L of length n is defined as PP21 n , where Pn1 is a path on n + 1 vertices, n ≥ 1. The graph obtained via this definition has the advantage of looking like a ladder having two rails and n + 1 rungs between them. The length of the ladder is defined as n.

Definition 3.2. A cycle-of-ladder is a graph containing a cycle Cb of length 2l called the bone

cycle and l ladders L1, L2, . . . , Ll with Rb(1), Rb(2), . . . , Rb(l) as the bottom rungs such that Rb(i)’s are respectively the alternate edges in Cb, 1 ≤ i ≤ l. We denote the cycle- of-ladder as CL(2l, s), where l and s represent the number of ladders and the length of each ladder respectively. i For convenience we label the vertices of Li as x j where 0 ≤ j ≤ s and 1 ≤ i ≤ l in CL(2l, s).

Theorem 3.3. Let G be a cycle-of-ladder CL(2l, s), l, s ≥ 4. Then src(G) = l(s + 1). Proof. Consider G to be a cycle of ladder CL(2l, s) where l represents the numbers of ladders and

s represents the length of each ladder. By lemma 2.1, if Cb is a cycle of length 2l, then src(Cb) = l. Therefore the cycle of length 2l is colored with l colors. Consider the ladders L1, L2 . . . Ll. Color the horizontal edges of each ladder with the color of Rb (i) and color parallel vertical edges of each ladder with distinct colors with the parallel edges receiving the same color. Consider vertices u and v in G. Let us assume that one vertex u in any one of the ladder say L1.

Let v be another in other ladder say Ll. Any (u, v) - path is a geodesic rainbow path. This implies that src(G) = l(s + 1). See Figure 2.

Figure 2: Strong rainbow coloring of CL(8, 4)

We add l number of edges to CL(2l, s) to obtain a 3-regular graph and call it the extended cycle- of-ladder ECL(2l, s).

Definition 3.4. The extended cycle-of-ladder ECL(2l, s) is obtained from CL(2l, s) by ii adding edges between (,)xxjj1 , 1 ≤ j ≤ s −1, the numbers taken modulo 2l.

Theorem 3.5. Let G be a cycle-of-ladder CL(2l, s), l, s ≥ 4. Then src(G) = l(s + 1). Proof. Consider G to be a cycle of ladder CL(2l, s) where l represents the numbers of ladders and s represents the length of each ladder. By lemma 2.1, if Cb is a cycle of length 2l, then src(Cb) = l. Therefore the cycle of length 2l is colored with l colors. Consider the ladders L1, L2 . . . Ll.

Color the horizontal edges of each ladder with the color of Rb(i) and color the parallel vertical edges of each ladder with distinct colors, with the parallel edges receiving the same color. Color ii the edges between ( ,xx )jj1 , 1 ≤ j ≤ s −1, as the color of the edge in Cb which is parallel to it. Consider the vertices u and v in G. Let us assume that one vertex u in any one of the ladder say L1. Let v be another in other ladder say Ll. Any (u, v) - path is a geodesic rainbow path. This implies that src(G) = l(s + 1). See Figure 3.

Figure 3: Strong rainbow coloring of ECL(8, 4)

4. Honeycomb Rhombic Torus

Definition 4.1. [7] Assume that m and n are positive integers where n is even. The honeycomb rhombic torus HRoT (m, n) is the graph with the node set {(i, j)\0 ≤ i < m, 0 ≤ j −i < n} such that (i, j) and (k, l) are adjacent if they satisfy one of the following conditions: 1. i = k and j = l ± 1(modn); 2. j = l and k = i − 1 if i + j is even; and 3. i = 0, k = m − 1, and l = j + m if j is even. In the graphical representation shown in Figure 4, the honeycomb rhombic torus comprises of horizontal, vertical, wraparound and oblique edges which are denoted by h, v, w and o respectively.

In HRoT(m, n), there are m paths of length n - 1, each of whose edges is a vertical edge. We call such a path as a column. Any two consecutive columns induce a ladder with horizontal edges as rungs, joining alternate vertices in the column.

Figure 4: Graphical Representation of HRoT(m, n)

Theorem 3.7. Let G be a honeycomb rhombic torus HRoT(m, n) where m and n are positive nm integers where n is even. Then src(G) =  2 . 2 Proof. Color the consecutive vertical edges in column i beginning from the top (1)(1)ininin 1,2,... and sequentially repeating till all the edges in the column are 222 colored, 1 ≤ i ≤ m. Color the rungs between column i and i + 1 with the colors in ini 1 1....n in the reverse order beginning from the top rung to the bottom rung. Color all 22 nm the wraparound edges with the color 1. Color the oblique edges with the color . 2 Geodesic rainbow paths of G are of the following categories. 1.The edges are distinct members of the set X = {h, v, w, o}. 2. One edge is v, the other two are h and one is wraparound or oblique edge. 3. One edge is h, the other two are v and one is wraparound or oblique edge. 4. One edge is v, one is h and one is either wraparound or oblique edge. In all the four cases there exists at least one geodesic rainbow path with distinct colors. Therefore src(G) = .

Figure 5: Strong rainbow coloring of honeycomb rhombic torus HRoT(4, 6). 5. Conclusion

In this paper the strong rainbow connection number are obtained for some special classes of graphs like ladder-like networks.

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