Abstract
Introduction:
Radio labeling of graphs extends the channel assignment problem by assigning non-negative integers to vertices of a connected graph G such that |h(℘)−h(𝓆)|≥diam(ℊ)+1−d(℘, 𝓆). The objective is to minimize the span, leading to the radio number rn(G).
Methods:
We consider a class of outerplanar graphs with vertex set {u1, v1, x1, …, xn, y1, …, yn} and a structured edge set combining path and matching edges. Analytical bounds and a constructive labeling algorithm are developed.
Results:
Lower and upper bounds for rn(G) are derived, and the proposed algorithm yields feasible radio labelings with near-optimal span.
Discussion:
The results highlight that the structure of outerplanar graphs enables efficient labeling strategies and provide a basis for extending radio labeling techniques to broader graph classes.
1 Introduction
In recent years, Hale's T-coloring problem (Hale, 1980) has become an important approach to address the channel assignment challenge. This problem involves assigning channels, denoted by non-negative integers, to individual radio transmitters. The aim is to allocate channels to any two transmitters that could interfere with each other and to ensure they are separated by a specific minimum distance. This channel assignment is significant in radio communication because interference could disrupt communication. Several studies have examined this area, including Tesman (1991); Liu (1996); Mari and Jeyaraj (2024a); Furedi et al. (1989); Mari and Jeyaraj (2024b); Chartrand and Zhang (2007); Mari and Jeyaraj (2023); Li et al. (2025); Cui and Li (2024), 2025), and Wang et al. (2026).
Liu (1996) introduced a variation of the Channel Assignment Problem (CAP), in which “nearby” transmitters are required to receive different channels. Furthermore, “extremely close” transmitters must be assigned channels with at least a difference of two. In this study, each transmitter is represented as a vertex. The relationship between two transmitters is defined as follows: if they are adjacent vertices in the graph, they are considered “very close,” while if they are at a distance of two vertices, they are classified as “close.” This approach is efficient for analyzing the relationships and proximity of transmitters in a given network.
Griggs and Yeh (1992) developed the distance-two labeling problem. |h(℘)−h(𝓆)|≥2 for vertices ℘ and 𝓆 at distance 1. Further, a k−L(2, 1)-labeling redefined by applying the condition that no label can exceed k. The L(2, 1)-labeling number of G, denoted as λ(G), shows the minimum value of k that allows for a valid k−L(2, 1)-labeling of the graph. Later, they calculated λ(Pn), λ(Cn), and λ(Wn), where Pn denotes a path with n vertices, Cn signifies a cycle with n vertices, and Wn is an n-wheel formed by adding a new vertex that connects to all vertices in Cn. For more detailed discussion, refer to the surveys by Calamoneri (2011) and Yeh (2006).
Chartrand and Erwin (2001) and Chartrand et al. (2005), introduced the concept of radio labeling. A graph G is labeled by a function h:V(G) → {0, 1, 2, 3, …} that satisfy the condition |h(℘)−h(𝓆)|≥diam(G)+1−d(℘, 𝓆). The label given to vertex ℘ by h is expressed as h(℘), while the span of the labeling is defined as span(h) = max{|h(℘)−h(𝓆)|:℘, 𝓆∈V(G)}. The radio number of G, represented as rn(G), is the minimum span obtained across all possible labelings h of the graph. A radio labeling for G with span equal to rn(G) is called an optimal radio labeling.
Radio labeling is one of the categories within the broader framework of L(℘1, ℘2, …, ℘d)-labeling problems (see Griggs and Yeh, 1992; Kral, 2006 and Zhou, 2008). . The objective is to minimize the span of a labeling function h:V(G) → {0, 1, 2, …} such that |h(℘)−h(𝓆)|≥rα whenever d(h(℘), h(𝓆)) = α, 1 ≤ α ≤ d. In this case, where d = diam(G) and rα = d−α+1 for all α, an L(d, d−1, …, 2, 1)-labeling becomes equivalent to a radio labeling.
This study provides a detailed analysis of the radio number of outerplanar graphs (OPGn). Our analysis indicates that the order of OPGn equals 2n+2. This study presents Theorems 1 and 2, which provide upper and lower bounds for the radio number of OPGn, respectively. The analysis concludes with Theorem 3, which asserts that
2 Some existing results
After reviewing relevant literature on radio labeling, we noticed that only two categories of graph labeling problems have been explored thus far. First, researchers have investigated methods for obtaining upper and lower bounds on the radio number of a graph. Second, they aim to fully determine the radio number of a graph. Significant advancements in these fields have been made, including the research by Liu and Zhu (2005), who investigate the radio number for both paths and cycles. Kchikech et al. (2007) provided exact values for the linear radio k-labeling number of stars and established an upper bound for trees. Chavez et al. (2021); Kchikech et al. (2008) further examined radio k-labeling in the context of Cartesian products of graphs, whereas Bantva (2017) concentrated on the radio number associated with the middle graph of paths. Chavez et al. (2021) analyzed radio k-labeling specifically for trees, and Khennoufa and Togni (2011) utilized a generalized approach to determine the radio antipodal number and the radio number of hypercubes. Li et al. (2010) provided enhanced lower bounds for certain tree structures and discussed radio numbers in relation to trees. Furthermore, Liu and Xie (2009) conducted an analysis of the radio number for square paths Liu, 2008).
3 Notation
In this section, a summary of the symbols and abbreviations used throughout this study is provided in Table 1.
Table 1
| Parameters | Description |
|---|---|
| OPGn or G | Outer planar graph with n vertices |
| u, v, xα, yα | Vertices of OPG |
| z | Number of vertices in OPG |
| C(OPG) | Center of OPG |
| d or diam(OPG) | Diameter of OPG |
| ε(v) | Eccentricity of a vertex v |
| ℓ(u) and L(OPG) | Level function and total level function |
| rn(OPGn) | Radio number of OPGn |
Notations used in the manuscript.
4 Preliminaries
In this section, the basic definitions used in the study are presented (for reference Harary, 1969 and Gallian, 2024). In a graph G, the length of the shortest path between ℘ and 𝓆 is denoted by d(℘, 𝓆). It is defined as
Eccentricity denotes the greatest distance from a vertex (℘) to any other vertex in G. Here, eccentricity is represented by ε(℘). A subgraph is defined here as the vertex whose eccentricity is lowest, and which is the center of the graph G, called C(G).
Definition 1. Let n ≤ 2 be an integer. Define OPG as the graph characterized by the vertex set {u, v, x1, x2, ..., xn, y1, y2, ..., yn} and the edge set given by {xαxα+1, yαyα+1:1 ≤ α ≤ n−1} ∪ {ux1, uy1, vxn, vyn} ∪ {xαyα:1 ≤ α ≤ n}. Examples are shown in Figures 1, 2
Figure 1
Figure 2
Lemma 1. For an OPG,
Proof. Let G be an outerplanar graph, and let S represent the set of vertices in G that have minimum eccentricity. The eccentricity of a vertex xα and yα if 1 ≤ α ≤ n is given by:
Case 1:nis odd. The values of ε(u) and ε(v) are n+1.
It is observable that within set S, both x(n+1)/2 and y(n+1)/2 are included. Consequently, in accordance with the provided function, both x(n+1)/2 and y(n+1)/2 have a smaller value than the eccentricity of the other vertices in the set OPG.
Case 2:nis even. The values are given by ε(u) and ε(v) is n+1.
One can see that . Therefore, from Case 1 and Case 2, we obtain
Let us now proceed to examine the new notation of the vertex of the OPG in relation to its center. The vertices of OPG will be renamed accordingly to reflect this new convention. This modification improves the clarity and consistency of the notation used in our analysis of the properties of OPG.
If n is even,
If n is odd,
Illustration: In Figures 3, 4, the outerplanar graph OPG7 and OPG8 are shown with new notation of vertices.
Figure 3
Figure 4
The level function concerning an OPG is defined on the set of whole numbers (W) by ℓ(℘) = min{d(℘, c):c∈V(C(OPG))}, for each (℘∈V(OPG))−V(C(OPG))). In OPG, the highest level is for odd n and for even n. Here, L(OPG), the total level function, is defined as follows:
5 Main results
In this section, the radio number of the outerplanar graph is determined. The outerplanar graph helps in a better understanding of the structure and its related radio properties. The algorithms have been developed, and subsequently, justified methods have been established for obtaining radio labeling of outerplanar graphs.
Observation:
|V(OPG))| = 2(n+1).
If ℘α, ℘α+1∈V(OPG), where 0 ≤ α ≤ z−2 are located on opposite sides and d(℘α, ℘α+1) is equal to d(℘α+1, ℘α+2), or d(℘α, ℘α+1) is equal to d(℘α+1, ℘α+2) ±1, or d(℘α, ℘α+1) is equal to d(℘α+1, ℘α+2) ±3, or d(℘α, ℘α+1) is equal to d(℘α+1, ℘α+2) ±4
Lemma 2. Let h be a function that assigns unique non-negative integers to V(OPG), and let ℘1, ℘2, ℘3, …, ℘z denote the sequence of V(OPG) such that h(℘α) < h(℘α+1), where h(℘1) = 0 and h(℘α+1) = h(℘α)+d+1−d(℘α, ℘α+1). Then, h is a radio labeling for 1 ≤ α ≤ z−1 whenever the following conditions hold:
d(℘α, ℘α+1) ≤ k+1 when n is odd.
d(℘α, ℘α+1) ≤ k+1 and d(℘α, ℘α+1) ≠ d(℘α+1, ℘α+2) when n is even.
Proof. Let h(℘1) = 0, and for each 1 ≤ α ≤ z−1, let h(℘α+1)≥h(℘α)+d+1−d(℘α, ℘α+1). Define hα as h(℘α+1)−h(℘α) for each 1 ≤ α ≤ z−1. Both of these conditions must be satisfied to confirm that h is a radio labeling. We need to show that for any α≠β, the inequality |h(℘α)−h(℘β)|≥d+1−d(℘α, ℘β) holds.
For simplicity, assume that β≥α+2 for each α = 1, 2, …, z−2. Then,
Case 1: The relationship defined in (1) is applied, when n is an odd value represented by 2k+1, and d = 2k, β≥α+2. Then,
Case 2: When n takes an even value of 2k, and d = 2k+1, we use the relation defined in (2), ensuring that β≥α+2, then
Given that α−β≠0, it can be observed that h is radio labeling in both cases.
Theorem 1. Let OPGn be a outerplanar graph on n vertices (n≥2). Then
Proof. To demonstrate the result, two cases are considered.
Case (1): When n = 2k. For OPG2k, let be defined by h(℘1) = 0,
According to the following ordering of vertices. We first set
and for 2 ≤ α ≤ z−1, define ℘h: = yα, where
Also define ℘h: = u1 and ℘h: = v1, where
Further, define ℘h: = xα, where
Case (2): When n = 2k+1. For OPG2k+1, define the labeling function by first assigning h(℘1) = 0, and for α = 1, 2, …, z−2,
and
We assign ℘1 = xk+1 and ℘z = yk+1. Observe that the order of the graph is z = 2(2k+1)+2 = 4k+4. For 2 ≤ α ≤ z−1, define the ordering as follows. First define ℘h: = yα, where
Similarly, define ℘h: = xα, where
Also define ℘h: = u1 and ℘h: = v1, where
By Lemma (2), the labeling h satisfies the radio condition for all consecutive and non-consecutive vertex pairs. Hence, h is a valid radio labeling of OPGn. Therefore,
Theorem 2. Let OPGn be a outerplanar graph on n vertices (n≥2). Then
Proof. Let h be an assignment of distinct non-negative integers to the vertices of OPG, where n≥2. Let ℘1, ℘2, ℘3, ..., ℘z denote the ordered sequence of vertices in V(OPG) such that h(℘α) < h(℘α+1). For OPG, define the function, h:V(OPG) → {0, 1, 2, ..., } as follows: h(℘α+1) = h(℘α)+d+1−(ℓ(℘α)+ℓ(℘α+1)) for α = 1, 2, ..., z−2 andh(℘z) = h(℘z−1)+d+1−(ℓ(℘z−1)+ℓ(℘z)).
Let h represent a radio labeling for OPG with a linear ordering of the vertices given by ℘1, ℘2, ℘3, ..., ℘z, such that h(℘1) = 0 < h(℘2) < h(℘3) < h(℘4) < ... < h(℘z). Then, for all ∀ 1 ≤ α ≤ z-1, it follows that h(℘α+1)−h(℘α)≥d+1−d(℘α, ℘α+1). By applying these z-1 inequalities, we obtain
Case 1: When n = 2k. For OPG2k, we have z = 4k+2, d = 2k+1, and . Since
Substituting into inequality (4.1), we obtain
Thus,
After simplification, we get
Case 2: When n = 2k+1. For the graph OPG2k+1, we have z = 4k+4, d = 2k+2, and . Hence, Substituting into inequality (Equation 9), to obtain
Thus,
After simplification, to obtain
Theorem 3. Let OPGn be a outerplanar graph on n vertices (n≥2). Then
Proof. The proof relies on Theorems (1) and (2), respectively.
Example 1. Figure 5 below illustrates the optimal radio labeling for OPGn = 7, along with the arrangement of the vertices.
Figure 5
xC0→vR→yL1→xR3→yL2→xR2→yL3→xR1→uL→yR3→xL1→yR2→xL2→yR1→xL3→yC0.
Example 2. Figure 6 below illustrates the optimal radio labeling for OPGn = 8, along with the arrangement of the vertices.
Figure 6
xL0→yR3→xL1→yR2→xL2→yR1→xL3→yR0→uL→vR→yL0→xR3→yL1→xR2→yL2→xR1→yL3→xR0.
6 Algorithm
In this section, we discuss the algorithm for radio labeling of outerplanar graphs. We have already discussed radio labeling of outerplanar graphs.
Understanding the Problem: We are tasked with developing a radio-labeling algorithm for a specific type of outerplanar graph, denoted as OPG. The structure of the graph is clearly defined, and we have a formula for the maximum label required based on the graph's length (n). Given a graph G(V, E), assign non-negative integers to its vertices. The label of a vertex u, denoted as h(u). We aimed to assign non-negative integer labels to vertices of a graph G such that: The difference in labels between any two vertices is at least diam(G)+1−d(℘, 𝓆), where diam(G) is the graph's diameter and d(℘, 𝓆) is the distance between vertices ℘ and 𝓆. The goal is to minimize the span, which is the maximum difference between any two labels.
A step-by-step procedure for the proposed algorithm is given below, and a detailed algorithm is mentioned in Algorithm 1.
1. Input:
Integer n representing the length of the OPG.
Number of vertices |V(OPG)| = 2n+2.
2. Output:
A labeling function h:V(G) → {0, 1, 2, 3, ..., } satisfies the radio labeling conditions.
3. Initialization:
Create an empty graph G and a list of unlabeled vertices.
Create vertex sets: U = u1, V = v1, X = {x1, x2, ...xn}, Y = {y1, y1, ..., yn}.
Create edge sets: E1 = (xα, xα+1), (yα, yα+1):1 ≤ α ≤ n−1, E2 = {u1x1, u1y1, v1xn, v1yn}, E3 = {xα, xα+1c; 1 ≤ α ≤ n}.
Combine edge sets: E = E1∪E2∪E3 .
Construct graph G with vertex set U∪V∪X∪Y and edge set E.
Assign label 0 to an arbitrary vertex and remove it from the unlabeled list.
4. Label Assignment:
Sort all vertices in non−decreasing order of their eccentricities. This helps in prioritizing central vertices.
Start with Central Vertex: Assign the label 0 to the vertex with the smallest eccentricity (most central).
Examine the adjacent vertices in proximity and ascertain their respective values.
The weights of the vertices, denoted as pα and pi+1, may be selected under the condition that adjacent vertices do not have the same weight. The weight of a vertex, denoted as h(a), is determined. The formula provided above is applied to each vertex within the graph, and this iterative process continues until all vertices have been processed.
Iterate until reaching a state of minimum span.
Start by labeling the vertices p1 and pz since they are central to the structure of the graph.
Assign p1 the label h(p1) = 0 and pz the label h(pz) = rn(OPG).
Algorithm 1
Input: The graph G = OPG. Output: Radio labelled graph G = OPG 1: procedure Algorithm RLOPG(a) 2: for Each vertex in OPG(V, E) do 3: Construct outerplanar graph(OPG) by applying 1. 4: Identifying the center of the vertices. 5: end for 6: for Each vertex(℘) in OPG do 7: Estimate the eccentrycity of (ε(℘) = max{d(℘, 𝓆)}:𝓆∈V(OPG) vertices. Applying 1 Case 1 and Case 2. 8: end for 9: for Assign center of vertices ←ε(℘) = rad(OPG) do 10: Identify the level function based on the center of the vertices. 11: for Each vertex(℘) in OPG(V) do 12: end for 13: ℓ(℘)←min{d(℘, 𝓆);𝓆∈V(C(OPG))} for each ℘∈V(OPG))−V(C(OPG)) 14: end for 15: Rename vertices labels using Equations 1–8. 16: for Each vertices ℘∈V(OPG) do 17: if h(℘α) < h(℘α+1) then 18: Assign distict non negative integer to V(OPG) based on Lemma 4.2 19: end if 20: end for 21: Radio labeling procedure 22: for Each vertices ℘∈V(OPG) do 23: if |h(℘)−h(𝓆)|≥diam(G)+1−d(℘, 𝓆). then 24: Assign h←{0, 1, 2, 3, ..., } 25: end if 26: end for 27: for Each vertices ℘∈V(OPG) do 28: if span(h) = max{|h(℘)−h(𝓆)|:℘, 𝓆∈V(G)}. then 29: rn(OPG)←minspan(k) 30: end if 31: end for 32: Return radio labeling graph G. 33: end procedure
7 Applications of outerplanar graphs
In this section, we present applications associated with outerplanar graphs (refer to Table 2). We aimed to explore radio labeling for outerplanar graphs.
Table 2
| Application | Description |
|---|---|
| Frequency assignment | Assigns unique frequencies to nodes to minimize interference, ensuring that adjacent nodes have different labels. |
| Network design | Helps design efficient wireless networks by ensuring optimal frequency distribution in outer planar topologies. |
| Interference reduction | Reduces the risk of communication interference in dense networks by utilizing the properties of outer planar graphs. |
| Dynamic spectrum management | Facilitates dynamic assignment of frequencies based on network conditions, adapting to changes in node connectivity. |
| Routing protocols | Enhances routing efficiency by managing labels (frequencies) to reduce collisions during data transmission. |
| Broadcasting | Enables efficient broadcasting by assigning frequencies in a way that avoids conflicts during simultaneous transmissions. |
| Distributed algorithms | Supports distributed frequency assignment algorithms that can efficiently operate in outer planar graph structures. |
| Resource optimization | Optimizes the use of limited frequency resources by leveraging graph properties to reduce redundancy and improve communication range. |
| Future research directions | Encourages exploration of new algorithms and strategies for radio labeling in varying network conditions. |
Applications of radio labeling in OPG.
The application lies in the field of wireless communication. Over time, the development of wireless networks has significantly improved communication between devices such as computers and telephones, enabling communication without relying on physical connections. However, the radio frequencies available for wireless communications are limited (for instance, in the U.S., all WiFi communications on the 2.4 GHz band are restricted to 11 channels). It is crucial to ensure secure transmission in areas such as cellular telephony, WiFi, security systems, and more (Molisch, 2012). A common issue is interference, such as when a phone call is disrupted by another user on the same line. This is caused by uncontrolled simultaneous transmissions, leading to interference. Channels that are too close can interfere or resonate, compromising communication. To prevent such interference, proper channel assignment is essential. Hale (1980) approached this issue by modeling it as a graph (vertex) coloring problem, known today as L(2, 1)-coloring. The aim is to minimize interference by assigning channels to a set of transmitters or stations. If transmitters or their frequency channels are physically close, interference can degrade the quality of time-sensitive or error-prone communications. To reduce interference, nearby transmitters must be assigned distinctly different channels.
Example: Consider a scenario with 16 transmitters, where the number of transmitters is given by V = {V1, V2, …, V16}. The communication network of these transmitters is represented as an outerplanar graph. Based on the conclusions drawn from Theorem 3, the optimal channel assignment for this network can be identified, as shown in Figure 5. In this case, the minimum optimal wireless labeling for the network is determined to be 62.
Figure 7 compares the radio number of OPG with several famous graph families. Here S1, 3⊗Pn signifies the Cartesian product produced by the star graph S1, 3 and the path graph Pn. The term Pn(P2) denotes the subdivision graph of the path Pn created by replacing each edge of Pn with a copy of P2, while M(Pn) refers to the middle graph of the path Pn. The comparison depicted in Figure 7 demonstrates the increase of the radio number for OPG compared to other graph families, specifically S1, 3⊗Pn, Pn(P2), and M(Pn). It is evident that the radio number of OPG grows in a systematic quadratic pattern as n increases. Compared with the other graph types, OPG shows a balanced growth rate, suggesting that its structure effectively distributes label distances while adhering to the radio labeling condition.
Figure 7
This behavior is significant for wireless communication applications. A controlled increase in radio number implies predictable channel separation requirements, which helps in designing networks where interference must be minimized. Hence, the obtained labeling scheme ensures stable channel allocation while maintaining efficient utilization of available frequencies.
8 Conclusions
The radio labeling problem is a widely applicable problem with significant real−world implications. While efficient algorithms for a small number of graph structures are available, exact labeling that can be measured by polynomial time is always desirable for any graph, regardless of its complexity. In this research, we present a complete determination of the radio number of outerplanar graphs and propose an algorithm based on outerplanar graph theory, represented as G = (OPGn). Our results make a significant contribution to understanding radio labeling problems and their applications across various domains.
Statements
Data availability statement
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.
Author contributions
BM: Formal analysis, Writing – original draft, Conceptualization, Methodology, Validation. RJ: Conceptualization, Validation, Supervision, Writing – review & editing.
Funding
The author(s) declared that financial support was not received for this work and/or its publication.
Conflict of interest
The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Generative AI statement
The author(s) declared that generative AI was not used in the creation of this manuscript.
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Summary
Keywords
channel assignment, graph labeling, outerplanar graph, radio labeling, radio number
Citation
Mari B and Jeyaraj RS (2026) Algorithm-based radio labeling for optimal channel assignment in outerplanar graphs. Front. Comput. Sci. 8:1748038. doi: 10.3389/fcomp.2026.1748038
Received
17 November 2025
Revised
25 February 2026
Accepted
01 April 2026
Published
29 April 2026
Volume
8 - 2026
Edited by
Erik Cuevas, University of Guadalajara, Mexico
Reviewed by
Feng Li, Qinghai Normal University, China
P. K. Niranjan, R. V. College of Engineering (RVCE), India
Updates
Copyright
© 2026 Mari and Jeyaraj.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Ravi Sankar Jeyaraj, ravisankar.j@vit.ac.in
Disclaimer
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