Open Neighborhood Coloring of Graphs - Semantic Scholar

Int. J. Contemp. Math. Sciences, Vol. 8, 2013, no. 14, 675 - 686 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijcms.2013.3337

Open Neighborhood Coloring of Graphs Geetha K. N. Department of Mathematics, Amrita Vishwa Vidyapeetham Bangalore, Karnataka State, India, Pin 560 035 K. N. Meera Department of Mathematics, Amrita Vishwa Vidyapeetham Bangalore, Karnataka State, India, Pin 560 035 Narahari N. Department of Mathematics, Tumkur University Tumkur, Karnataka State, India, Pin 572 101 B. Sooryanarayana Department of Mathematical and Computational Studies Dr.Ambedkar Institute of Technology Bangalore, Karnataka State, India, Pin 560 056 c 2013 Geetha K. N. et al. This is an open access article distributed under Copyright the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract For a simple, connected, undirected graph G(V, E) an open neighborhood coloring of the graph G is a mapping f : V (G) → Z + such that for each w ∈ V , and ∀u, v ∈ N (w), f (u) = f (v). The maximum value of f (w), ∀w ∈ V (G) is called the span of the open neighborhood coloring f . The minimum value of span of f over all open neighborhood colorings f is called open neighborhood chromatic number of G, denoted by χonc (G). In this paper we determine the open neighborhood chromatic number of some standard graphs, trees and the infinite triangular lattice.

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Geetha K. N., K. N. Meera, Narahari N. and B. Sooryanarayana

Mathematics Subject Classification: 05C15 Keywords: Neighbor, open neighborhood, coloring

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Introduction

All the graphs considered here are undirected, connected and simple. We use standard terminologies and the terms not defined here may be found in [2, 7, 6]. A k-coloring of a graph G is a coloring f : V (G) → S, where S ⊂ Z + and |S| = k. The elements of S are called colors. A k − coloring is proper if every pair of adjacent vertices receives different colors. A graph is k − colorable if it has a proper k − coloring. The chromatic number χ(G) is the least k such that G is k − colorable. For results on proper coloring, we refer [6]. We define open neighborhood coloring of a graph G(V, E) as a coloring f : V (G) → Z + , such that for each w ∈ V (G) and ∀u, v ∈ N(w), f (u) = f (v). The maximum value of f (w), ∀w ∈ V (G) is called the span of the open neighborhood coloring f . The minimum value of span of f over all open neighborhood colorings f is called open neighborhood chromatic number of G, denoted by χonc (G).

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Figure 1: A proper coloring and an open neighborhood coloring of the graph G Remark 1.1. An open neighborhood coloring of a graph G need not be a proper coloring of G. The definition of open neighborhood coloring of a graph G is motivated by the definition of L(h, k) labeling [14]. Given a graph G(V, E), and two non-negative integers h and k, an L(h, k) labeling is an assignment of non

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Open neighborhood coloring of graphs

negative integers to the vertices of G such that adjacent vertices are labeled using colors at least h apart, and vertices having a common neighbor are labeled using colors at least k apart. This definition imposes a condition on labels of vertices connected by a 2-length path instead of using concept of distance 2, which is very common in the literature. When h ≥ k, the two definitions coincide, but are different when h < k. For example when h < k, the former definition of L(h, k) labeling requires that the vertices of a triangle be differently labeled where as the latter definition does not as shown in the Figure 2 and 3. Most of the work in literature deals with the latter definition

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Figure 3: An L(0,1) labeling where verFigure 2: An L(0,1) labeling where vertices connected by a path of length 2 are

tices at distance 2 are labeled differently.

labeled differently.

of L(h, k) labeling with h ≥ k [3, 4, 8, 11, 10, 5, 13]. The L(h, k) labeling problem is studied mainly to avoid hidden terminal interference in multihop radio networks in [1] and for code assignment in computer networks in [9]. Our definition of open neighborhood coloring is found to be equivalent to the former definition of L(h, k) labeling dealing with the case of h < k, choosing h = 0 and k = 1, which has been rarely dealt with. The L(0, 1) labeling of a special class of graphs called cactus graphs has been studied in [12] . In this paper, we determine the open neighborhood chromatic number of some standard graphs and the triangular lattice which is an infinite graph. By the definition of open neighborhood coloring, the coloring starts from 1 instead of 0.

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Some preliminary results

Observation 2.1. By the definition it follows that if f is an open neighborhood coloring of G(V, E) with span(f ) = χonc (G), then f (u) = f (v) holds if and only if the vertices u, v are the end vertices of a path of length 2 in G. Observation 2.2. For any graph G(V, E), χonc (G) ≥ Δ(G). Observation 2.3. If H is a connected subgraph of G, then χonc (H) ≤ χonc (G). Observation 2.4. The open neighborhood chromatic number of a connected graph G, is 1 if and only if G ∼ = K1 or K2 .

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Theorem 2.5. If G χonc (G) ≤ n − 1.

is a triangle free graph with n vertices then

Proof. We prove the theorem in two cases. Case 1: Suppose there exists a vertex v of degree n − 1. Then G ∼ = K1,n−1 as G is triangle free. In this case χonc (G) ≥ n − 1. Further assigning one color to each of the n − 1 vertices adjacent to v and repeating one of these colors to v, gives an open neighborhood coloring of G. Thus χonc (G) = n − 1. Case 2: Suppose there exists no vertex v of degree n − 1. Then every vertex is adjacent to at most n − 2 vertices of G. In this case assigning one color each to the n − 2 vertices adjacent to any vertex and assigning the (n − 1)th color to the vertex gives an open neighborhood coloring of G. Thus χonc (G) ≤ n − 1.

Theorem 2.6. Let G be any graph and v0 be a vertex of maximum degree Δ(G). Suppose H is a subgraph induced by N[v0 ] such that every v ∈ N(v0 ) belongs to a triangle containing v0 . Then, χonc (G) ≥ Δ(G) + 1. Proof. By Observation 2.2, χonc (G) ≥ Δ(G). Further v0 cannot be assigned any of the colors assigned to its neighbors v, as v0 and v belong to some triangle in H and hence have a common neighbor. Therefore χonc (H) ≥ deg(v0) + 1 = Δ(G) + 1. By Observation 2.3, χonc (G) ≥ χonc (H). Hence χonc (G) ≥ χonc (H) ≥ Δ(G) + 1. Theorem 2.7. Let G(V, E) be any graph and G1 (V, E ) be the graph obtained by adding an edge between every pair of vertices u and v of G that are connected by a path of length two. Then, χonc (G) ≥ χ(G1 ). Proof. In an open neighborhood coloring of a graph G we assign different colors to the vertices u and v which are connected by a path of length two in G. Since these two are adjacent in G1 an open neighborhood coloring of G is nothing but a proper coloring of G1 and χ(G1 ) is the minimum number of colors required for a proper coloring of G1 . Hence the result. ¯ is Theorem 2.8. If G is a graph of diameter 2 whose complement G ¯ connected then, χonc (G) ≥ χ(G). Proof. Let u, v ∈ V (G) such that dG (u, v) = 2. Then there exists a path of length 2 between them in G. So any open neighborhood coloring of G assigns different colors to u and v. Now dG¯ (u, v) = 1 and hence an open neighborhood ¯ But χ(G) ¯ is the minimum number of coloring of G is a proper coloring of G. ¯ Hence χonc (G) ≥ χ(G). ¯ colors required for a proper coloring of G.

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3

Open Neighborhood standard Graphs

Coloring

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Theorem 3.1. If G ∼ = Kn , n ≥ 3 or K1 + Pn−1 , n ≥ 3 or W1,n−1 , n ≥ 4 then χonc (G) = |V (G)|. Proof. Every pair of vertices in G, are the end vertices of some path of length 2 in G. By the Observation 2.1, every pair of vertices in G must be given a different color. Hence the result. Theorem 3.2. For a path Pn , n ≥ 2, 1, if n = 2, χonc (Pn ) = 2, if n ≥ 3. Proof. Let Pn be a path on n ≥ 2 vertices with V (Pn ) = {v1 , v2 . . . vn }. The case of n = 2, has been dealt in Observation 2.4. We now consider the case of n ≥ 3. As Δ(Pn ) = 2, by Observation 2.2 we have χonc (Pn ) ≥ 2. We now show that χonc (Pn ) = 2, by defining the coloring f : V (Pn ) → Z + as 1, if i ≡ 1, 2 (mod 4) f (vi ) = 2, if i ≡ 0, 3 (mod 4). Now for all i, 2 ≤ i ≤ n − 1, N(vi ) = {vi−1 , vi+1 }. If i ≡ 0 (mod 4) then f (vi−1 ) = 2 = 1 = f (vi+1 ). If i ≡ 1 (mod 4) then f (vi−1 ) = 2 = 1 = f (vi+1 ). If i ≡ 2 (mod 4) then f (vi−1 ) = 1 = 2 = f (vi+1 ). If i ≡ 3 (mod 4) then f (vi−1 ) = 1 = 2 = f (vi+1 ). Therefore f (vi−1 ) = f (vi+1 ), ∀i, 2 ≤ i ≤ n − 1. Hence χonc (Pn ) = 2 when n ≥ 3. Theorem 3.3. For a cycle Cn , n ≥ 3, 2, if n ≡ 0 (mod 4), χonc (Cn ) = 3, otherwise. Proof. Let Cn be a cycle on n ≥ 3 vertices with V (Cn ) = {v0 , v1 . . . vn−1 }. The case of n = 3, has been dealt in Theorem 3.1. We now consider the case of n ≥ 4 in four cases. As Δ(Cn ) = 2, by Observation 2.2 we have χonc (Cn ) ≥ 2. Case(1): n ≡ 0 (mod 4) For all i, 0 ≤ i ≤ n − 1, we define the coloring f : V (Cn ) → Z + as

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f (vi ) =

1, if i ≡ 0, 1 (mod 4) 2, if i ≡ 2, 3 (mod 4).

Now for all i, 0 ≤ i ≤ n − 1, N(vi ) = {vi−1 , vi+1 }, where the suffix is under modulo n. If i ≡ 0 (mod 4) then f (vi−1 ) = 2 = 1 = f (vi+1 ). If i ≡ 1 (mod 4) then f (vi−1 ) = 1 = 2 = f (vi+1 ). If i ≡ 2 (mod 4) then f (vi−1 ) = 1 = 2 = f (vi+1 ). If i ≡ 3 (mod 4) then f (vi−1 ) = 2 = 1 = f (vi+1 ). Therefore for all i, 0 ≤ i ≤ n − 1, f (vi−1 ) = f (vi+1 ) . Hence χonc (Cn ) = 2 when n ≥ 4, n ≡ 0 (mod 4). Let Cn (V , E ) be the graph with V = V (Cn ) and E = {vi vj / vi and vj are end vertices of a path of length 2 in Cn }. Then by Theorem 2.7, χonc (Cn ) ≥ χ(Cn ). We now consider the cases of n ≡ 1, 2, 3 (mod 4) and show in each case that χ(Cn ) = 3 and hence χonc (Cn ) ≥ 3. Case(2): n ≡ 1 (mod 4) For Cn , n ≡ 1 (mod 4), Cn (V , E ) is an odd cycle with V = V (Cn ) and E = {v0 v2 , v2 v4 , v4 v6 . . . vn−1 v1 , v1 v3 , . . . vn−2 v0 } and any odd cycle requires three colors for a proper coloring. Therefore χ(Cn ) = 3 and hence χonc (Cn ) ≥ 3. We show that χonc (Cn ) = 3. For all i, 0 ≤ i ≤ n − 2, we define the coloring f : V (Cn ) → Z + as 1, if i ≡ 0, 1 (mod 4) f (vi ) = 2, if i ≡ 2, 3 (mod 4) and f (vn−1 ) = 3 Now for all i, 0 ≤ i ≤ n − 1, N(vi ) = {vi−1 , vi+1 }, where the suffix is under modulo n. For all i, 1 ≤ i ≤ n − 3, If i ≡ 0 (mod 4) then f (vi−1 ) = 2 = 1 = f (vi+1 ). If i ≡ 1 (mod 4) then f (vi−1 ) = 1 = 2 = f (vi+1 ). If i ≡ 2 (mod 4) then f (vi−1 ) = 1 = 2 = f (vi+1 ). If i ≡ 3 (mod 4) then f (vi−1 ) = 2 = 1 = f (vi+1 ).


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Also, N(v0 ) = {v1 , vn−1 } and f (v1 ) = 1 = 3 = f (vn−1 ). N(vn−2 ) = {vn−3 , vn−1 } and n ≡ 1 (mod 4) ⇒ f (vn−3 ) = 2 = 1 = f (vn−1 ). N(vn−1 ) = {vn−2 , v0 } and n ≡ 1 (mod 4) ⇒ f (vn−2 ) = 2 = 1 = f (v0 ). Therefore for all i, 0 ≤ i ≤ n − 1, f (vi−1 ) = f (vi+1 ) . Hence χonc (Cn ) = 3 when n ≥ 4 and n ≡ 1 (mod 4). Case(3): n ≡ 2 (mod 4) In this case Cn (V , E ) is a disjoint union of two odd cycles, Cn 1 (V1 , E1 ) and Cn 2 (V2 , E2 ) where V1 = {v0 , v2 . . . vn−2 }, E1 = {v0 v2 , v2 v4 . . . vn−2 v0 } and V2 = {v1 , v3 . . . vn−1 }, E2 = {v1 v3 , v3 v5 . . . vn−1 v1 }. As Cn 1 (V1 , E1 ) is an odd cycle χ(Cn 1 ) = 3. The same colors can be used to properly color the odd cycle Cn 2 (V2 , E2 ) as the two cycles are disjoint. Hence χ(Cn ) = 3 and by Theorem 2.7 χonc (Cn ) ≥ χ(Cn ) = 3. We now show that χonc (Cn ) = 3. For all i, 0 ≤ i ≤ n − 3, we define the coloring f : V (Cn ) → Z + as 1, if i ≡ 0, 1 (mod 4) f (vi ) = 2, if i ≡ 2, 3 (mod 4). and f (vn−2 ) = f (vn−1 ) = 3 Now for all i, 0 ≤ i ≤ n − 1, N(vi ) = {vi−1 , vi+1 }, where the suffix is under modulo n. For all i, 1 ≤ i ≤ n − 4, If i ≡ 0 (mod 4) then f (vi−1 ) = 2 = 1 = f (vi+1 ). If i ≡ 1 (mod 4) then f (vi−1 ) = 1 = 2 = f (vi+1 ). If i ≡ 2 (mod 4) then f (vi−1 ) = 1 = 2 = f (vi+1 ). If i ≡ 3 (mod 4) then f (vi−1 ) = 2 = 1 = f (vi+1 ). Also, N(v0 ) = {v1 , vn−1 } and f (v1 ) = 1 = 3 = f (vn−1 ). N(vn−3 ) = {vn−4 , vn−2 } and n ≡ 2 (mod 4) ⇒ f (vn−4 ) = 2 = 1 = f (vn−2 ). N(vn−2 ) = {vn−3 , vn−1 } and n ≡ 2 (mod 4) ⇒ f (vn−3 ) = 2 = 1 = f (vn−1 ). N(vn−1 ) = {vn−2 , v0 } and n ≡ 1 (mod 4) ⇒ f (vn−2 ) = 2 = 1 = f (v0 ). Therefore for all i, 0 ≤ i ≤ n − 1, f (vi−1 ) = f (vi+1 ) . Hence χonc (Cn ) = 3 when n ≥ 4 and n ≡ 2 (mod 4). Case(4) : n ≡ 3 (mod 4) For Cn , n ≡ 3 (mod 4), Cn (V , E ) is an odd cycle with V = V (Cn ) and E = {v0 v2 , v2 v4 , v4 v6 . . . vn−1 v1 , v1 v3 , . . . vn−2 v0 } and any odd cycle requires three colors for a proper coloring. Therefore χ(Cn ) = 3 and hence by Theorem 2.7 χonc (Cn ) ≥ 3. We now show that χonc (Cn ) = 3. For all i, 1 ≤ i ≤ n − 1, we define the coloring

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f (vi ) =

1, if i ≡ 0, 1 (mod 4) 2, if i ≡ 2, 3 (mod 4).

and f (v0 ) = 3 Now for all i, 1 ≤ i ≤ n − 1, N(vi ) = {vi−1 , vi+1 }, where the suffix is under modulo n. For all i, 2 ≤ i ≤ n − 2 If i ≡ 0 (mod 4) then f (vi−1 ) = 2 = 1 = f (vi+1 ). If i ≡ 1 (mod 4) then f (vi−1 ) = 1 = 2 = f (vi+1 ). If i ≡ 2 (mod 4) then f (vi−1 ) = 1 = 2 = f (vi+1 ). If i ≡ 3 (mod 4) then f (vi−1 ) = 2 = 1 = f (vi+1 ). Also, N(v0 ) = {vn−1 , v1 } and n ≡ 3 (mod 4) ⇒ f (vn−1 ) = 2 = 1 = f (v1 ). N(v1 ) = {v0 , v2 } and f (v0 ) = 1 = 2 = f (v2 ). Further, N(vn−1 ) = {vn−2 , v0 } and n ≡ 3 (mod 4) ⇒ f (vn−2 ) = 1 = 3 = f (v0 ). Therefore for all i, 0 ≤ i ≤ n − 1, f (vi−1 ) = f (vi+1 ) . Hence χonc (Cn ) = 3 when n ≥ 4 and n ≡ 3 (mod 4). Hence the result. Theorem 3.4. For a complete bipartite graph Km,n where m, n ≥ 2, χonc (Km,n ) = max{m, n}. Proof. Let V (Km,n ) = V1 V2 , with V1 = {v1 , v2 , . . . vm } and V2 = {u1 , u2, . . . un }, where m, n ≥ 2 Without loss of generality, we may assume m = max{m, n}. (The proof for the case of n = max{m, n} is similar). Therefore, Δ(Km,n ) = m. By Observation 2.2, we have, χonc (Km,n ) ≥ m. We now prove that χonc (Km,n ) = m by defining the coloring f : V (Km,n ) → Z + as follows : For all i, 1 ≤ i ≤ m, f (vi ) = i and for all j, 1 ≤ j ≤ n, f (uj ) = j. Now, for all i, 1 ≤ i ≤ m, N(vi ) = {u1 , u2 . . . un } and f (ul ) = l = k = f (uk ) for all 1 ≤ l, k ≤ n. Also, for all j, 1 ≤ j ≤ n, N(ui ) = {v1 , v2 . . . vm } and f (vl ) = l = k = f (vk ) for all 1 ≤ l, k ≤ m. Therefore, f is an open neighborhood coloring which uses a minimum of m colors as m = max{m, n}. Hence the result. Theorem 3.5. For a tree T on n ≥ 3 vertices, χonc (T ) = Δ(T ). Proof. By Observation 2.2, we have χonc (T ) ≥ Δ(T ). We now show that Δ(T ) colors are sufficient for an open neighborhood coloring of T . Let v0 be


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the vertex of maximum degree in T . Let deg(v0) = Δ(T ) = k. Consider the tree to be rooted at v0 . Let N(v0 ) = {v1 , v2 . . . vk }. These k vertices must be assigned k different colors say c1 , c2 , . . . ck . Any one of these colors say c1 may be assigned to v0 . Now consideran arbitrary vertex say vi ∈ N(v0 ). Now deg(vi ) ≤ k. As T is a tree, N(v0 ) N(vi ) = ∅. Let S = N(vi ) − v0 . Thus |S| ≤ k − 1. Any vj ∈ S is connected to v0 by a path of length 2 and hence cannot be assigned the color c1 . Hence we are left with k − 1 colors which can be reused to color the vertices of the set S. Hence k = Δ(T ) colors are sufficient for an open neighborhood coloring of T . Theorem 3.6. Let G(V, E) bea connected graph on n ≥ 3 vertices. Then χonc (G) = n if and only if N(u) N(v) = ∅ holds for every pair of vertices u, v ∈ V (G). Proof.Let u, v ∈ V (G) be any two arbitrary vertices of G such that N(u) N(v) = φ. ⇒ There exists w ∈ V (G), such that w ∈ N(u) N(v). ⇒ w ∈ N(u) and w ∈ N(v) ⇒ u ∈ N(w) and v ∈ N(w) ⇒ f (u) = f (v). Hence every pair of vertices in G are assigned different colors. Therefore χonc (G) = n. We now prove the converse part. Suppose N(u) N(v) = φ holds for some pair of vertices u, v ∈ V (G). ⇒ there exists no vertex w ∈ V (G) such that w ∈ N(u) N(v) ⇒ there exists no vertex w ∈ V (G) such that w ∈ N(u) and w ∈ N(v). ⇒ there exists no vertex w ∈ V (G) such that u, v ∈ N(w). ⇒ there exists an open neighborhood coloring f : V (G) → Z + with f (u) = f (v). ⇒ χonc (G) ≤ n − 1 a contradiction. Hence the result.

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Open neighborhood coloring of a triangular lattice

In this section we determine the open neighborhood chromatic number of the infinite triangular lattice which is defined as follows : Let ε1 = (1, 0), ε2 = √ 1 3 ( 2 , 2 ) be two vectors in Euclidean plane. The triangular lattice is the infinite graph ΓΔ with vertex set V (ΓΔ ) = {iε1 + jε2 : i, j ∈ Z} and E(ΓΔ ) = {uv : u, v ∈ V (ΓΔ ), dE (u, v) = 1}, where dE (u, v) is the euclidean distance between u and v. In short we denote the vertices by (i, j).

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The triangular lattice plays an important role in radio broadcasting and mobile cellular networks. In a mobile radio network, large service areas are often covered by a network of congruent polygonal cells, with each transmitter in the center of the cell that it covers. A honeycomb of hexagonal cells provides the most economic covering and here the transmitters are placed in a triangular lattice.

Figure 4: Triangular lattice ΛΔ

Figure 5: < v

N1 (v)

N2 (v) >

Theorem 4.1. The open neighborhood coloring of the triangular lattice is 7.

Proof. For any vertex v ∈ V (ΓΔ ), let N1 (v) be the neighbors of v1 and N2 (v) be the set of vertices that are connected to v by a path of length 2. Then, for v = (i, j) ∈ V (ΓΔ ), N1 ((i, j)) = { (i ± 1, j), (i, j ± 1), (i ± 1, j ∓ 1)} and N2 ((i, j)) = { (i±2, j), (i, j ±2), (i±1, j ∓2), (i±2, j ∓2), (i±1, j ±1), (i± 2, j ∓ 1)}. The Figure 5 shows the induced subgraph of v N1 (v) N2 (v). The induced subgraph of v N1 (v), is a wheel W1,6 . Then by Theorem 3.1, χonc (W1,6 ) = 7. Also by Observation 2.3 χonc (ΓΔ ) ≥ χonc (W1,6 ) = 7. We now show that χonc (ΓΔ ) = 7.

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Define the coloring f : V (ΓΔ ) → Z + as : ⎧ (i + 3)(mod 7), if i ≡ 0, 1, 2, 3, 5, 6 (mod 7), j ≡ 0(mod ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 7, if i ≡ 4 (mod 7), j ≡ 0(mod 7) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (i + 1)(mod 7), if i ≡ 0, 1, 2, 3, 4, 5 (mod 7), j ≡ 1(mod ⎪ ⎪ ⎪ ⎪ ⎪ 7, if i ≡ 6 (mod 7), j ≡ 1(mod 7) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (i − 1)(mod 7), if i ≡ 0, 2, 3, 4, 5, 6 (mod 7), j ≡ 2(mod ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 7, if i ≡ 1 (mod 7), j ≡ 2(mod 7) ⎪ ⎪ ⎪ ⎪ ⎨ (i − 3)(mod 7), if i ≡ 0, 1, 2, 4, 5, 6 (mod 7), j ≡ 3(mod f (i, j) = ⎪ 7, if i ≡ 3 (mod 7), j ≡ 3(mod 7) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (i + 2)(mod 7), if i ≡ 0, 1, 2, 3, 4, 6 (mod 7), j ≡ 4(mod ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 7, if i ≡ 5 (mod 7), j ≡ 4(mod 7) ⎪ ⎪ ⎪ ⎪ ⎪ i (mod 7), if i ≡ 1, 2, 3, 4, 5, 6 (mod 7), j ≡ 5(mod ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 7, if i ≡ 0 (mod 7), j ≡ 5(mod 7) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (i − 2)(mod 7), if i ≡ 0, 1, 3, 4, 5, 6 (mod 7), j ≡ 6(mod ⎪ ⎪ ⎪ ⎩ 7, if i ≡ 2 (mod 7), j ≡ 6(mod 7)

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The vertices N1 (v) and N2 (v) are the only vertices connected to v by a path of length 2. By the coloring defined above none of these vertices receive the color of v, as shown in the Figures 6 and 7 when f (v) = 1 and f (v) = 2 respectively. A similar coloring can be obtained when f (v) is assigned any color from 3 through 7 by cyclically changing the colors 1 to 2, 2 to 3 and so on up to 7 to 1. In all these colorings the largest color used is 7. Hence χonc (ΓΔ ) = 7

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References [1] A. A. Bertossi and M.A. Bonuccelli, Code assignment for hidden terminal interference avoidance in multihop packet radio networks, IEEE/ACM Trans Networking 3 (1995), 441-449. [2] Buckley F and Harary F, Distance in Graphs, Addison-Wesley,(1990). [3] David Kuo and Jing-Ho Yan,On L(2,1)labelings of carteisan product of paths and cycles, Discrete Mathematics, 283(2004), 137-144. [4] Gerard J. Chang and David Kuo, The L(2,1)labeling problem on graphs SIAM Journal of Discrete Mathematics, Vol 9. No. 2, pp 309-316, May 1996. [5] Gerard J. Chang et al, On L(d,1)labelings graphs , Discrete Mathematics, 220(2000), 57-66. [6] F. Harary, Graph theory, Narosa Publishing House, New Delhi, 1969. [7] Hartsfield Gerhard and Ringel, Pearls in Graph Theory, Academic Press, USA, 1994. [8] Jerrold R. Griggs and Roger. K. Yeh, Labeling graphs with a condition at distance two, SIAM Journal of Discrete Mathematics, Vol 5, No. 4, pp 586-595, Nov 1992. [9] X. T. Jin and R. K. Yeh,Graph Distance-Dependent Labeling related to code assignment in computer networks, Wiley Interscience Vol 52 (2005), 159-163. [10] John P. Georges, David W. Mauro and Melanie I. Stein, Labeling products of complete graphs with a condition at distance two, SIAM Journal of Discrete Mathematics, Vol 14. No. 1, pp 28-35. [11] John P. Georges, and David W. Mauro, On generalized petersen graphs labeled with a condition at distance two, Discrete Mathematics, 259(2002), 311-318. [12] Nasreen Khan,Madhumangal Pal and Anita Pal, On L(0,1)- Labelling of Cactus Graphs, Communications and Network, Vol 4, 18-29,2012. [13] Roger. K. Yeh, A survey on labeling graphs with a condition at distance two, Discrete Mathematics, 306(2006), 1217-1231. [14] Tiziana Calamoneri The L(h,k)-labeling problem: A survey and annotated bibiliography , The Computer Journal, 49 (2006), 585-608. Received: March 16, 2013