Radial radio number of chess board graph and king’s graph

A radial radio labeling ד of a connected graph G = (V,E) with radius rad(G) is a mapping from V (G) to N ∪ {0} satisfying |ד(u)−ד(w)|+ d(u,w) ≥ 1 + rad(G), ∀ u, v ∈ V (G). The span of a radial radio labeling ד, denoted by rr(ד) is the greatest number in the range of ד. The minimum span taken over all radial radio labelings ד of G is called the radial radio nmber of G and it is denoted by rr(G). In this article, we have investigated the upper bounds for rr(G) of chess board graphs and king’s graph.

INTRODUCTION In today's digital age, our world has been transformed into a higher dimension by digital technology, especially in the field of communication technology. In the beginning of the 21 st century, Chartrand et al. [1] were motivated by the maximum channel allocation in a fixed spectrum introduced the concept of radio k-chromatic number in graph theory. For any k lies between 1 and the diameter of G, the radio k− chromatic number is defined as follows: Let G = (V, E) be connected graph and d be its diameter, then a radio kcoloring ℸ of graph G is an assignment of V (G) to non-negative integers such that |ℸ(u) − ℸ(w)|+ d(u, w) ≥ 1 + k, ∀ u, w ∈ V (G), where d(u, w) is the distance between u and w in G. The span of a radio k− chromatic number of h denoted by r ck (ℸ) is the largest number in the range of ℸ. The radio k− chromatic number of G is the minimum value taken over all such radio k -chromatic number of ℸ. For different k values, different names were given to the radio k− chromatic number by the researchers in the recent research articles. Namely, when the k values 1 and 2, then the problem is called the chromatic number and L(2, 1) labeling number respectively. Chang and Kue [2] introduced the L(2, 1)−labeling of graphs. Hasunuma et al. [3] derived a linear time algorithm for L(2, 1) labeling of trees. Yenoke et al. [4], [5] determined the upper bound for the L(2, 1) problem of slim tree, comb graph, double comb graph and Christmas tree, silicate and oxide networks  [6] obtained the L(2, 1) labeling number of crown graph and line graph of armed crown graph. Smitha and Thirusangu [7] determined the L(2, 1) labeling of cycle related graphs. Besides, L(2, 1) labeling of unigraph was computed by Calamoneri and Petreschi [8].
If we vary the value of k as 3, then the problem is named as L(3, 2, 1) labelling. Xavier et al. [9] obtained the bounds for the L(3, 2, 1) labelling number of for both n-star graph S(n, r) and n-star-wheel graph SW (n, r) as 2r + 2n + 1 and n-wheel star graph W S(n, r) as 3r + 2n. Kim et al. [10] solved the L(3, 2, 1) labeling problem the product of C n and K n . Moreover, Amanathulla and Pal [11] studied the same on interval graphs. Further, when k reaches the maximum value d, this problem insistence the assigning of channels to FM radio stations called radio labeling problem which was introduced by Chartrand [12]. Rajan et al. [13] and Rajan and Yenoke [14] obtained the lower bound for the radio number of any connected graph and also investigated the exact radio number of wheel graph W n+1 , double fan graph DF n , fan graph F n , windmill graph K m n , star graph S n+1 and uniform r -cyclic split graph KC(r) as n + 2, n + 3, n + 2, m(n − 1) + 2, n + 3 and nkr + 3n − 2 respectively. Vaidya and Bantva [15] attained the exact radio number of the total graph of path P n for n = 2k and n = 2k + 1 as 4k(k − 1) + 2 and 4k 2 + 3 respectively. Recently, Yenoke and Kaabar [16] investigated the bounds for the Nanostar tree dendrimer T n,p (n, p > 2) as rn(T n,p ) ≤ n + (2n − 1)p + 1 respectively. In addition, when k = d − 1, it was named as antipodal radio number by [17], [18]. William and Kenneth [19] investigated the bounds for the antipodal radio number of lobster graph as an(L(m, r, k)) ≤ 24r + 20k − 7. Saha and Panigrahi [20] studied the same problem for some powers of cycles. Avadayappan et al. [21] were introduced the radial radio labeling by fixing the k value as the radius of the graph.
A radial radio labeling ℸ of a connected graph G = (V, E) with radius rad(G) is a mapping from The span of a radial radio labeling ℸ, denoted by rr(ℸ) is the greatest number in the range of ℸ. The minimum span taken over all radial radio labelings ℸ of G is called the radial radio nmber of G and it is denoted by rr(G). This problem is very helpful in dividing a network into sub networks and to apply the radio labeling conditions in assigning the channels for a particular divided geographical area. Especially, if there is a need of partitioning the existing network into two sub networks, this labeling technique can be applied without affecting the optimal channel assignment.
Recently, Jose and Giridharan [24] proved that rr(M T (n)) ≤ 2n + 1 and rr(D(n)) ≤ 2n + 2, where M T (n) and D(n) are Mongolian tent and diamond graphs respectively. In this paper we have estimated the bounds for the radial radio number of certain interconnection networks such as chess board graphs and King's graph.

DEFINITIONS AND TERMINOLOGY
In this section we have listed few definitions and results which will be used for proving the theorems. Let u be a vertex of a connected graph G, then the eccentricity of u denoted e(u) is the farthest vertex from u to any other vertex v in G. That is, e(v) = max{d(u, v)∀v ∈ V (G)}. The radius of the graph G is the minimum TELKOMNIKA Telecommun Comput El Control ❒ 11 eccentricity of the vertices of G and it is denoted by rad(G). Pardalos et al. [25] defined the following chess board and its related graphs. An m × n chessboard graph denoted by CB(m, n) is defined as the Cartesian product P m × P n of paths on m and n vertices respectively. In the literature it is also denoted by m × n mesh. The mn vertices in CB(m, n) are named as {(k, l)\l = 1, 2 . . . m, k = 1, 2 . . . n}. If m = n, then the radius of CB(m, n) is 2 n 2 . A 2 × n chessboard graph with 2n vertices is also called a ladder graph denoted by L n . The radius of L n is n 2 + 1. An m × n King's graph denoted by KG(m, n) is a graph which is obtained by all legal moves of the king chess piece on a m × n chessboard CB(m, n). More specifically, it is constructed by the strong product of the paths P m and P n . The radius of KG(m, n) is n 2 .

MAIN RESULTS
In this section we have obtained the bounds the radial radio number of 2 × n and n× n chessboard graph separately. Further, we have determined the bounds for the radial radio number of n × n king's graph. a) Theorem 1: Let L n be a ladder graph with 2n vertices, then the radial radio number of L n satisfies Proof: We prove this theorem using two cases based on the value of n, odd or even. Define a radial radio labeling from the vertex set of L n to the non-negative integers separately for odd and even cases as follows: 1) Case 1: n is even.
Define a labeling from the vertex set of L n to the non-negative integers as follows:  Since n is even, the radius of L n is n 2 +1. Therefore, we must prove that |ℸ(u)−ℸ(w)|+d(u, w) ≥ n 2 + 2 for all u, w ∈ V (L n ) 1.1) Case 1.1: Suppose u ̸ = w belongs to the upper row of L n , then the following three sub cases can arise.
Hence, we substitute the values of i and j as n+1 . c) Theorem 3: For n > 2, the radial radio number of n×n chessboard graph CB(n, n) satisfies rr(CB(n, n)) ≤ n 2 (n+3) 4 , whenever n is even.

ISSN: 1693-6930
See the Figure 4(b). The rest of the proof is omitted.
(a) (b) Figure 4. A radial radio labeling of KG(n,n) for (a) n = 6 and (b) n = 7 which illustrates the mapping

CONCLUSION
The upper bounds for the radial radio number of chess board graphs CB(m, n) for m = 2, n and the King's graph KG(m, n) for m = n (n > 2) has been investigated in this research work. For m ̸ = n (n > 2), CB(m, n) and KG(m, n) is still an open problem. Future this research can be extended to identify higher dimensional networks and study the same radial radio number problem due to its application to telecommunication networks.