Optimal radio labelings of graphs

Let $\mathbb{N}$ be the set of positive integers. A radio labeling of a graph $G$ is a mapping $\varphi : V(G) \rightarrow \mathbb{N} \cup \{0\}$ such that the inequality $|\varphi(u)-\varphi(v)| \geq diam(G) + 1 - d(u,v)$ holds for every pair of distinct vertices $u,v$ of $G$, where $diam(G)$ and $d(u,v)$ are the diameter of $G$ and distance between $u$ and $v$ in $G$, respectively. The radio number $rn(G)$ of $G$ is the smallest number $k$ such that $G$ has radio labeling $\varphi$ with $\max\{\varphi(v) : v \in V(G)\}$ = $k$. Das et al. [Discrete Math. $\mathbf{340}$(2017) 855-861] gave a technique to find a lower bound for the radio number of graphs. In [Algorithms and Discrete Applied Mathematics: CALDAM 2019, Lecture Notes in Computer Science $\mathbf{11394}$, springer, Cham, 2019, 161-173], Bantva modified this technique for finding an improved lower bound on the radio number of graphs and gave a necessary and sufficient condition to achieve the improved lower bound. In this paper, one more useful necessary and sufficient condition to achieve the improved lower bound for the radio number of graphs is given. Using this result, the radio number of the Cartesian product of a path and a wheel graphs is determined.


Introduction
The channel assignment problem is the problem to assign channels to each TV or radio transmitters such that the interference constraints are satisfied and the use of spectrum is minimized.
The problem was first introduced by Hale [11] in 1980. The interference between transmitters is closely related to geographic location of the transmitters. The closer the transmitters are, the higher the interference is and vice-versa. Hence, the frequency difference between two radio channels assigned to radio transmitters is in the inverse proportion to the distance between two transmitters. Initially only two level interference, namely high and low, was considered and accordingly, two transmitters are called very close and close, respectively. In a private communication with Griggs during 1988, Robert proposed a variation of the channel assignment problem in which close transmitters must receive different channels and very close transmitters must receive channels that are at least two apart. The problem is studied by mathematicians using graphs labeling approach.
In a graph, the transmitters are represented by vertices and two vertices are adjacent if two transmitters are very close and distance two apart if they are close. The problem of assignment of channels to transmitters is associated with graph labeling problem. Motivated through this problem, Griggs and Yeh introduced L(2, 1)-labeling (or distance two labeling) in [9] as follows: An L(2, 1)-labeling of a graph G = (V (G), E(G)) is a function ϕ from the vertex set V (G) to the set of non-negative integers such that |ϕ(u) − ϕ(v)| ≥ 2 if d(u, v) = 1 and |ϕ(u) − ϕ(v)| ≥ 1 if d(u, v) = 2, where d(u, v) is the distance between u and v in G. The span of ϕ is defined as span(ϕ) = max{|ϕ(u) − ϕ(v)| : u, v ∈ V (G)}. The λ-number, denoted by λ(G), is defined as the minimum span over all L(2, 1)-labelings of G. The L(2, 1)-labeling and other distance two labeling problems have been studied by many researchers in the past two and half decades; for example, see the survey articles [4,20].
In [5,6], Chanrtrand et al. extended the constraint on distance from two to the largest possible distance; the diameter of graph G and introduced the concept of radio labeling as follows: set of natural numbers) such that the following is satisfied for every pair of distinct vertices The assigned integer ϕ(u) is called the label of u under ϕ and, the span of ϕ is defined as The radio number of G, denoted by rn(G), is defined as with minimum taken over all radio labelings ϕ of G. A radio labeling ϕ is optimal if span(ϕ) = rn(G).
It is clear that an optimal radio labeling ϕ always assign 0 to some vertex and hence the span of ϕ is the maximum integer assigned by ϕ. A radio labeling is a one-to-one integral function from V (G) to the set of non-negative integers. Therefore, any radio labeling ϕ induces It is clear that if ϕ is an optimal radio labeling then span(ϕ) ≤ span(ψ) for any other radio labeling ψ of G.
A radio labeling problem is recognized as one of the tough graph labeling problems. In [5,6], Chartrand et al. gave an upper bound for the radio number of paths and cycles. Liu and Zhu determined the exact radio number for paths and cycles in [15]. Even determining the radio number for basic graph families like paths and cycles was challenging. In [16,17,18], Vaidya and Bantva determine the radio number for total graph of paths, strong product P 2 with P n and linear cacti. The radio number of trees remain the focus of many researchers in recent years. In [10], Halász and Tuza determine the radio number of level-wise regular trees. In [13], Li et al.
determine the radio number of complete m-ary trees. In [14], Liu gave a lower bound for the radio number of trees and, a necessary and sufficient condition to achieve the lower bound; the author presented a class of trees, namely spiders, achieving this lower bound. In [3], Bantva et al. gave a different necessary and sufficient condition to achieve this lower bound and presented banana trees, firecrackers trees and a special class of caterpillars achieving this lower bound.
necessary and sufficient condition to achieve the lower bound in [2]. They also discussed the radio number of line graph of trees and block graphs. Liu et al. also studied the radio k-labeling of trees in [7]. In [8], Das et al. gave a technique to find a lower bound for the radio number of graphs. In [1], Bantva improved this technique to find a lower bound for the radio number of graphs and gave a necessary and sufficient condition to achieve the improved lower bound.
Using these results, the author determined the radio number of the Cartesian product of paths and Peterson graph.
In this paper, one more useful necessary and sufficient condition to achieve the improved lower bound for the radio number of graphs given in [1] is established. Some subgraphs of a given graph G are characterized such that if the radio number of G achieves the lower bound given in [1] then these subgraphs also achieve the lower bound. Using these results, the radio number of the Cartesian product of the path graphs with wheel, star and the friendship graphs are determined.

Preliminaries
The book [19] is followed for standard graph theoretic terms and notation. Only simple finite connected graphs are considered throughout this paper. The distance d G (u, v) between two vertices u and v is the least length of a path joining u and v in a graph G. The suffix is dropped whenever the graph G is clear in the context. The diameter of a graph G, denoted by diam(G), The subgraph induced by S, denoted by G(S), is a subgraph of G whose vertex set is S and Let H be an induced connected subgraph of G with diam(H) = k. Define layers L i of graph G with respect to subgraph H as follows: Set L 0 = V (H) and L 1 = N (L 0 ). Recursively define which is known as the maximum level in a graph G. Since G is a connected graph, Define the total distance of layers of graph G, denoted by L(G), as For a graph G, define Let G be any connected graph then for any u, v ∈ V (G), note that the distance between u and v in a graph G satisfies the following inequality: In [8], Das et al. gave a technique to find a lower bound for the radio number of graphs.
In [1], Bantva improved this technique and gave a lower bound for the radio number of graphs which is given in the following theorem.
[1] Let G be a simple connected graph of order p, diameter d and L 0 ⊆ V (G).
Denote k = diam(L 0 ) and δ = δ(G). Then Although, both the lower bounds given in [8] and [1] seems to be identical in notation but the difference lies in fixing the L 0 . In [8], Das et al. set a vertex or a clique of graph G as L 0 while Bantva set all vertices of an induced subgraph H of G as L 0 with the property that two non-adjacent vertices of V (H) have distance equal to diam(L 0 ). The readers may notice that this improved technique gives a better lower bound for the radio number of graphs, which is sharp for some classes of graph. The author of [1] presented one such class of graphs, which consists of the Cartesian product of the path graph and Peterson graph in [1]. In this paper, the condition to fix L 0 is further relaxed as follows.
subgraph of G with the property that the vertices of G can be ordered as In [1], Bantva also gave a necessary and sufficient condition (given in the next theorem) to achieve the lower bound (3) for the radio number of graphs.

Main Result
In this Section, we give one more useful necessary and sufficient condition to achieve the improved lower bound for the radio number of graph given in [1], which rely only on the ordering of vertices of a graph.
Theorem 3.1. Let G be a simple connected graph of order p, diameter d ≥ 2 and L 0 is fixed in G as described earlier. Denote k = diam(L 0 ) and δ = δ(G). Then holds if and only if there exists an ordering O(V (G)) := (x 0 , x 1 , . . . , x p−1 ) of V (G) such that the following conditions are satisfy.
Moreover, under the conditions (a) and (b), the mapping ϕ defined by is an optimal radio labeling of G.
Proof. Necessity: Suppose that (5) Note that ϕ is a radio labeling of G and so ϕ( Sufficiency: Suppose that an ordering O(V (G)) := (x 0 , x 1 , . . . , x p−1 ) of V (G) satisfies conditions (a)-(b) of hypothesis and ϕ is defined by (7) and (8). Note that it is enough to prove that ϕ is a radio labeling with span equal to the right-hand side of (5). Let x i and x j (0 ≤ i < j ≤ p−1) be two arbitrary vertices then by (8) and using (6), we have and hence ϕ is a radio labeling. The span of ϕ is given by . This together with (3) implies (5).
A graph with no cycle is called acyclic graph. A forest is an acyclic graph. A tree is a connected acyclic graph. A spanning subgraph of a graph G is a subgraph with vertex set V (G).
Observe that the diameter of P m ✷W n is m + 1.
Proof. We consider the following two cases.
Case-1: m is even. In this case, set {(u m/2 , v 0 ), (u m/2+1 , v 0 )} of P m ✷W n as L 0 then diam(L 0 ) = k = 1 and the maximum level in P m ✷W n is h = m/2.
In this case, set {(u (m+1)/2 , v 0 )} of P m ✷W n as L 0 then diam(L 0 ) = k = 0 and the maximum level in P m ✷W n is h = (m + 1)/2. The order of P m ✷W n and L(P m ✷W n ) are given by Substituting (13) and (14) into (3) we obtain the right-hand side of (10) which is a lower bound for the radio number of rn(P m ✷W n ). We prove that this lower bound is tight. Let τ and σ are as defined earlier in Case-1. Let α be a permutation defined on {1, 2, . . . , n} as follows: Using permutations α, τ and σ, we first rename (u i , v j )(1 ≤ i ≤ m, 0 ≤ j ≤ n) as (a r , b s ) as follows: if 1 ≤ i ≤ m and j = 0; or i = m and 1 ≤ j ≤ n; if i = (m + 1)/2 and 1 ≤ j ≤ n; (u i , v στ (j) ), if 2 ≤ i ≤ (m − 1)/2 and 1 ≤ j ≤ n; (u i , v σ(j) ), if (m + 3)/2 ≤ i ≤ m − 1 and 1 ≤ j ≤ n.
Claim-2: The above defined ordering O(V (P m ✷W n )) := (x 0 , x 1 , . . . , x p−1 ) satisfies (6). Let x i and x j (0 ≤ i < j ≤ p − 1) be any two arbitrary vertices. Denote the right-hand side of (6) by . If b = a + 1 then we consider the following two cases: (i) j = i + 2n and (ii) j = i + 2n. If j = i + 2n then d(x i , x j ) = 1 and in this case, E(i, j) < 0 < d(x i , x j ) and if j = i + 2n then d(x i , x j ) = 2 and in this case, otherwise E(i, j) ≤ 2 ≤ d(x i , x j ) which completes the proof of Claim-2.
An n-star, denoted by K 1,n , is a tree consisting of n leaves and another vertex joined to all leaves by edges. Denote the vertex set of K 1,n by V (K 1,n ) = {v 0 , v 1 , . . . , v n } with E(K 1,n ) = {v 0 v i : 1 ≤ i ≤ n}. A friendship graph F n is a graph obtained by identifying one vertex of n copies of cycle C 3 with a common vertex. Denote the vertex set of F n by V (  Proof. Observe that P m ✷F n can be regarded as a subgraph of P m ✷W 2n with identical L 0 = {(u m/2 , v 0 ), (u m/2+1 , v 0 )} when m is even and L 0 = {(u (m+1)/2 , v 0 )} when m is odd and hence by Theorem 3.3, the radio number of P m ✷W 2n and P m ✷F n are identical.
Example 3.1. In Table 1, an ordering of vertices and the corresponding optimal radio labeling of P 7 ✷W 7 is shown. Table 1: An ordering and optimal radio labeling for vertices of P 7 ✷W 7 .
(u i , v j ) i→ Example 3.2. In Table 2, an ordering of vertices and the corresponding optimal radio labeling of P 8 ✷W 7 is shown.