Abstract
Communication systems including AM and FM radio stations transmitting signals are capable of generating interference due to unwanted radio frequency signals. To avoid such interferences and maximize the number of channels for a predefined spectrum bandwidth, the radio-k-chromatic number problem is introduced. Let be a connected graph with diameter and radius . For any integer , , radio coloring of G is an assignment of color (positive integer) to the vertices of such that , where is the distance between and in G. The biggest natural number in the range of is called the radio chromatic number of G, and it is symbolized by . The minimum number is taken over all such radio chromatic numbers of which is called the radio chromatic number, denoted by . For and , the radio chromatic numbers are termed as the radio number () and radial radio number () of , respectively. In this research work, the relationship between the radio number and radial radio number is studied for any connected graph. Then, several sunflower extended graphs are defined, and the upper bounds of the radio number and radial radio number are investigated for these graphs.
1. Introduction
The channel frequency assignment problem was first proposed by Griggs and Yeh [1] in 1992 for the amplitude modulation radio stations. Due to the cochannel interference, there is a challenge to fix the transmitters in a particular geographical area. Therefore, studying the channel assignment problem in radio stations is NP‐complete. However, Fotakis et al. [2] proved that even for graphs with diameter 2, the problem is NP-hard. Chartrand et al. [3] presented the theoretical graph definition for the radio-k-chromatic number as follows.
Let be a connected graph with diameter and radius . For any integer , , radio coloring of G is an assignment of color (positive integer) to the vertices of such that , where is the distance between and in G. The biggest natural number in the range of is called the radio chromatic number of G, and it is symbolized by . The minimum number is taken over all such radio chromatic numbers of which is called the radio chromatic number, denoted by .
Čada et al. [4] proved that, for any distance graph we have
Recently, Bantva [5] improved this general lower bound. Based on different values, the radio chromatic number is classified into different problems.
For , the radio chromatic number is termed as the radio number problem, and it is symbolized by . It was introduced by Chartrand et al. [6] for the purpose of determining the maximum number of channels for frequency modulation (FM) radio stations by minimum utilization of spectrum bandwidth. The radio number problem has been studied by several researchers [7, 8]. In 2017, Avadayappan et al. [9] brought in the concept of radial radio labelling. A mapping for a connected graph is called a radial radio labelling if this satisfies the inequality where is the radius of the graph . Radial radio number of symbolized by is the maximum number mapped under . The radial radio number of , denoted by , is equal to . A few number of research articles [10, 11] were published in the area of radial radio labelling. In this paper, we have studied a comparative relation between and . Furthermore, we have defined and determined the radio and radial radio numbers of certain sunflower extended graphs such as , and .
2. Relation between the Radio Number and Radial Radio Number
This section deals with certain results which connect with for any connected graph .
Definition 1. The eccentricity of a vertex z, represented by in a connected graph , is the maximum distance from to any other vertex in G. That is, . The maximum eccentricity of the vertices of G is called the diameter of the graph, and it is symbolized by d or . In addition, the radius of graph G, symbolized by or , is the minimum eccentricity of the vertices of G.
Definition 2. A connected graph is called a self-centred graph if In other words, .
The following is a straight result from the definitions of the radio number and radial radio number.
Theorem 1. For any connected graph ,.
Chartrand et al. [6] proved the following three theorems, which will be used to study the general results for the radial radio number.
Theorem 2. If is a connected graph of order and diameter , then .
Theorem 3. For a complete partite graph of order , .
Theorem 4. Every connected graph of order with is self-centred.
Using Theorem 5 and Definition 2, we have attained the equality of Theorem 1 as follows.
Theorem 5. A connected graph of order is self-centred if and only if .
Theorem 6. Let be a complete partite graph of order ; then, .
Proof. Let the vertex set of be partitioned into disjoint sets such that , and . The radius of the complete partite graph is 1, and all the vertices in the sets , are at distance two. Hence, we can label the vertices in each set as . Clearly, the radial radio labelling condition is satisfied for any pair of vertices in . Hence, .
Theorem 7. If is a connected graph of order and radius , then .
Proof. Given is a connected graph that contains at least two vertices. Therefore, the lower bound of the theorem attains in the particular case of Theorem 6 which is for the complete bipartite graphs. Furthermore, the upper bound is obtained by replacing by in Theorem 2. Consequently,
3. Results and Discussion
In this section, we have defined and investigated the radial radio and radio number of some sunflower extended graphs such as star-sun graph complete-sun graph wheel-sun graph , and fan-sun graph .
Definition 3. A sunflower graph consists of a wheel with a centre vertex , -cycle , and additional vertices where is joined with edges to (), , and is taken as modulo . It is represented by The radius, diameter, and number of vertices of are 2, 4, and , respectively.
Definition 4. A star graph, denoted by , is defined as a complete bipartite graph of the form . In other words, is a tree having leaves and one internal vertex.
Definition 5. A star-sun graph, denoted by , is a graph obtained from the sunflower graph and copies of star graph by merging the internal vertex of the star graph and vertex of , , as shown in Figure 1(a).
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Remark 1. The cardinality of and in is and , respectively. Also, the diameter and radius of the graph are 6 and 3, respectively.
Definition 6. A complete-sun graph, denoted by , is a graph obtained from the sunflower graph and copies of complete graph by merging a vertex of the complete graph and the vertex of , , as shown in Figure 1(b). Here, we have and .
Remark 2. The diameter and radius of are 6 and 3, respectively.
Definition 7. A wheel-sun graph, denoted by , is a graph obtained from the sunflower graph and copies of wheel graph by merging the vertex of and the centre vertex of the wheel, where as shown in Figure 1(c).
Remark 3. The number of vertices in is , while its number of edges is . Also, its diameter and radius are 6 and 3, respectively.
Definition 8. A fan-sun graph is a graph obtained from the sunflower graph and copies of fan graph by merging of the fan and the vertex of , . It is denoted by as shown in Figure 1(d).
Remark 4. For the graph the number of edges is , while the number of vertices is . Moreover, the diameter and radius are 6 and 3, respectively.
In this work, we name the newly included vertices of , , and as in the clockwise sense.
3.1. Radial Radio Number of Sunflower Extended Graphs
The following theorems provide the upper bound for the radial radio number of , .
Theorem 8. Let G be the sun-star graph Then, .
Proof. First, we define a mapping as follows: , , , , , and as shown in Figure 2.
Since the radius of the graph is 3, we must verify satisfies the radial radio labelling condition for every pair of vertices .
Let us choose any two arbitrary vertices and in the sun-star graph. Case 1: suppose a and b are star vertices, then and are of the form and Case 1.1: if , then the value of and under is and , respectively. Also, and are at a distance two. Hence, the radial radio labelling condition becomes since . Case 1.2: if , then and are at a distance at least 4. Hence, the radial radio labelling condition is trivially satisfied. Case 2: let , , and ; then, the value of is , and is at least . Furthermore, is at least 2. Therefore, . Case 3: if we take , , and , then since , which trivially verifies the radial radio labelling condition. Case 4: suppose , , and , then the value of is , and is at least . Furthermore, is at least 2. Therefore, . Case 5: if is the centre vertex of the wheel and is any other star vertex, then the distance between them is exactly 3. Also, and . Therefore, . Case 6: let and be the vertices in the -cycle of the sunflower graph. Case 6.1: if and , , then . Again, . Since , the condition for the radial radio labelling is satisfied. Case 6.2: suppose and are of the form and , where , then the distance between them is exactly two. Also, the function values of and are and , respectively. Hence, the radial radio labelling condition becomes . Case 6.3: suppose , , and , , then and .If and , then ; else, Hence, in both possibilities, we obtain and . Case 7: if , and , then is at least . Since , the radial radio labelling condition is easily verified. Case 8: suppose and . Case 8.1: if and , , respectively, then and . Also, the distance between and is . Hence, we have since . Case 8.2: if and , , then . Also, takes the values of and to and , respectively. Therefore, . Case 8.3: let , , and , ; then, and are mapped to and, respectively. When and , , and hence, . The remaining possibility is obvious since the distance between them is at least 1, and the modulus difference between and is at least 4. Case 9: if and is any vertex of the form , then the verification is obvious since the difference in the function values of and is at least Thus, is a valid radial radio labelling, and the vertex receives the maximum number . That is, Hence, we conclude that .
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Theorem 9. Let G be the complete-sun graph . If then the radial radio number of satisfies
Proof. Let us name the newly included vertices of as in the clockwise sense. Now, we define a one-one mapping as follows:This mapping is visible in Figure 3(a).
In the following, we claim that .
Let . Case 1: suppose and , , then and Case 1.1: if , then and Therefore, since . Case 1.2: if , then , which is enough for verifying the condition. Case 2: assume that , , , and ; then, and . Also, Therefore, . Case 3: if we take , , , and , then since , which verifies the condition trivially. Case 4: assume that and Case 4.1: if and , then and However, . Therefore, . Otherwise, is greater than 3. Case 5: let and , . In this case, if and or and or and , 0 then and Otherwise, and Hence, in both possibilities, the condition for radial radio labelling is satisfied. Case 6: assume that and , . If and or and , , , then . Thus, and Otherwise, and Therefore, the condition holds in both of the possibilities. Case 7: finally, let us assume that , and is any vertex in . If then and Otherwise, the radial radio labelling condition is obviously true. Thus, satisfies the condition of radial radio labelling and attains the maximum value for the vertex . Therefore, we get The proof for the other two cases, namely, and , is left to the reader.
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Theorem 10. For , the radial radio number of the wheel-star graph satisfiesWe omit the proof, but Figure 3(b) illustrates the case .
3.2. Radio Number of Sunflower Extended Graphs
This section provides the upper bound for the radio number of , .
Theorem 11. For and the radio number of the complete-sun graph satisfies
Proof. We define a 1-1 mapping as follows:See Figure 3(a).
Then, to show is a valid radio labelling, we must verify the inequalityLet . Case 1: suppose that and Case 1.1: if or or where and , then and . In the same subcase, if then and So, Case 1.2: if , , then . In addition, . Therefore, Case 1.3: if , , then Consequently, the condition is true. Case 1.4: if , , then and . It follows that . Case 2: take and If , , then and Otherwise, that is, if , thus, and Therefore, in both chances, we get . Case 3: assume that and then and ; else, and Therefore, in both of them, the inequality is Case 4: suppose that and , ; then, either and or and . Therefore, . Case 5: if we take , , and , , then from the mapping, , which verifies the condition trivially. Case 6: assume that and , . If and , then and Otherwise, So, Case 7: let be the centre vertex of the wheel and be any other vertex in the graph. If , then . Otherwise, the condition is obviously true. Thus, is a valid radio labelling, and the vertex is labelled with the maximum number . Hence, , and .
Theorem 12. For and the radio number of the star-sun graph satisfies .
Proof. First, we define an injection as follows:We omit the rest of the proof as it is similar to the one for Theorem 11.
The other two cases in Theorems 11 and 12 are also left to the reader.
Theorem 13. For , the radio number of the wheel-sun graph satisfiesFigure 4 illustrates the case . We omit the proof of this theorem.
4. Conclusion
In this paper, we have presented the relation between the radio number and radial radio number. We have also defined and investigated the bounds for the same problems for the graphs , . For the graph fan-sun graph the problem is still considered as an open research problem that needs further investigation. Since the method of finding the radial radio number and radio number of the fan-sun graph is similar to the previous theorem, it is still open to the interested researchers to do a further research work that can extend our results to identify more relations between the radio number and radial number by studying the same problem for interconnection networks.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
Mohammed K. A. Kaabar contributed to actualization and initial draft, provided the methodology, validated and investigated the study, supervised the original draft, and edited the article. Kins Yenoke validated and investigated the study, provided the methodology, performed formal analysis, and contributed to the initial draft. Both authors read and approved the final version.