RETRACTED ARTICLE: A study on harmonious chromatic number of total graph of central graph of generalized Petersen graph

Abstract

For a given graph $G$, $\chi_{h} \left( G \right)$ is the least integer of colors such that no two adjacent nodes receive the same color and each combination of color seems together on at most one line. The least number of compartments into which a warehouse should be partitioned to store chemicals certain pairs of which are incompatible is again the chromatic number of the conflict graph. In this article we have found the harmonious coloring of central graph of generalized Petersen graph and have characterized the harmonious chromatic number with the maximum matching number and further we have found the total graph of central graph of generalized Petersen graph using the clique. Clique is a complete graph where every vertex is adjacent to every other. In computational biology we use cliques as a technique of abstracting pair wise relationships consisting of protein–protein interaction or gene similarity. In the latter case we would need to set up a side among the vertices representing two genes if the ones genes have say comparable expression profiles over several time factors of an experiments to study the health care information of patients.

1 Background

Discrete Mathematics is that piece of science, which manages a deliberate treatment and comprehension of discrete structures and procedures experienced in our day by day life, which are regularly intrinsically very mind boggling in nature however apparently justifiable. This character of discrete mathematics is teeming with energy, once in a while even to a typical man. The labelling of discrete structures is likewise a field which has a similar trademark. The issues merging from the investigation of an assortment of labeling plans of the components of a graph, or of any discrete structure is a potential territory of research.

Graph theory is a prospering control containing a collection of excellent and incredible consequences of wide relevance. Its unstable development as of late is for the most part because of its job as a basic structure supporting current connected arithmetic—software engineering, combinatorial improvement, and tasks investigate specifically—yet additionally to its expanding application in the more connected sciences. Problems in Graph theory can be portrayed as presence issues (Königsberg bridge, four coloring and so forth.), development issues count issues and improvement issues. Graphs appear a naturally common approach to display numerous circumstances in the creation. This clarifies why Graph Theory has developed so significantly in the previous century. It has now settled itself as a control without anyone else.

Among the various types of problems that show up while examining Graph theory, one that has been becoming solid is the territory that reviews labelling of graphs, which is a part of research in combinatory. A graph labeling is a task of numbers to the vertices or edges, or both, subject to certain conditions. Graph coloring is an uncommon instance of graph labelling; it is a task of marks generally called hues to components of a graph subject to certain limitations. In its least complex structure, it is a method for coloring the vertices of a graph. Graph coloring has been contemplated as an algorithmic issue since the early 1970s: the chromatic number problem is one of Karp’s 21 NP- complete problems from 1972 and at around a similar time different exponential-time calculations were created dependent on backtracking and on the cancellation withdrawal.

A harmonious coloring of a simple graph is a proper coloring with the end goal that each pair of hues shows up together on at most one line. The harmonious chromatic number is the minimal number of hues in such a coloring. This parameter was presented by Miller and Pritikin. Each graph has a harmonious coloring, since it does the trick to allocate each vertex a different color. It was appeared by Hopcroft et al. (1983) that the problem of deciding the harmonious chromatic number of a graph is NP—hard, and a short evidence of a similar outcome, because of Johnson, shows up in a similar paper. The main paper on harmonious coloring was distributed by Frank et al. (1982). Be that as it may; the correct meaning of this thought is expected to (Hopcroft et al. 1983; Lee and Mitchum 1987) gave an upper bound to the harmonious chromatic number of graphs. The harmonious chromatic number problem was explored by Beane et al. Miller et al. McDiarmid et al. and Krasikov et al. Lu has gotten estimates for the harmonious chromatic number of a few classes of graphs. He has likewise decided limits on the harmonious chromatic number of a total paired and trinary tree. Later Lu gave the exact estimation of the harmonious chromatic number of a total paired tree and trinary tree. In 1995 Edwards researched the harmonious chromatic number problem for nearly all trees (Edwards 1995), and in 1997 he proceeded with his work on limited degree trees (Edwards 1997), limited degree graph and distributed papers relating harmonious chromatic number and the achromatic number. He underscored another upper destined for the harmonious chromatic number in the year 1998 (Edwards 1998), furthermore, on the harmonious chromatic number of complete r-ary trees in the year 1998. Mitchem (1989) has acquired different hypotheses on harmonious chromatic number furthermore, talked about different open inquiries. Another lower headed for the harmonious chromatic number was given by Campbell et al. as far as the independence number.

Further work on the harmonious coloring can be found in papers by Thilagavathi et al. (2009), Franklin Thamil Selvi and Amutha (2016) and Vivin et al. (2007) acquired outcomes on harmonious coloring of central graph of odd cycles and complete graphs and on the middle graph of central graph of cycle graphs and so on. They have likewise examined a similar issue for line graphs of central graphs of bipartite graphs. Bodlaender gives a proof that builds up the NP-completeness of the harmonious coloring problem for disconnected interval graphs and co graphs. Asdre et al. (2007) expanded Bodlaender’s outcomes by demonstrating that the issue remains NP-complete for connected interval graphs. Furthermore, the NP-completeness of the issue has been demonstrated for trees what's more, connected bipartite permutation graphs, connected quasi-threshold and threshold graphs. Since the issue of deciding the harmonious chromatic number of an associated cograph is inconsequential, Asdre et al. (2007) demonstrated that the harmonious coloring problem is polynomially reasonable on associated semi limit and edge graphs. Now it merits seeing that the issue is much harder when limited to many chart families in which the difficult issues typically become tractable. There are just a couple of families for which we can have accurate arrangements in polynomial time.

In healthcare systems, it is essential to study the gene patterns (Zhu et al. 2019) to estimate the hereditary diseases and malnourishment deficiency. The computational biology plays a vital role in estimating the links between gene patterns and similarity between the genes (Han et al. 2019). The computation involves forming links to the similarities and form the vertices as central graph. Harmonious Chromatic Number of central graphs determines the maximum matching number and minimum edge cover of Generalized Petersen models thus the weak links identified to determine the deficiency in gene pattern.

Fig. 1

$\chi _{h} [GP(5,2)] = 10$

2 Preliminary

Definition 2.1

Given a simple graph, $\chi_{h} \left( G \right)$is the least integer of colors in G such that no two adjacent nodes receive the same color and each combination of color seems together on at most one line Fig. 1.

Definition 2.2

Given a simple graph $G$, $C\left(G\right)$ is acquired by dividing every line of $G$ precisely once henceforth attaching remaining non contiguous nodes of $G$ Fig. 2.

Fig. 2

C[GP(3,1)]

Definition 2.3

For natural numbers $n$ and $\left(n>2k\right)$, $GP\left(n,k\right)$, is interpreted by node set $\left\{{u}_{i}\right.,\left.{v}_{i}\right\}$, and line set $\left\{{u}_{i}{u}_{i+1}\right.,\left.{{u}_{i}v}_{i},{u}_{i}{v}_{i+k}\right\}$; where $i=\mathrm{1,2},\dots \dots .,n$ and subscripts are reduced modulo $n$. We call two nodes $u$ and $v$ the twin of each other and refer to the edge between them as a spoke. Hence the number of nodes and lines are $2n$ and $3n$ respectively Fig. 3.

Fig. 3

GP(8,3)

Definition 2.4

The total graph of $G$ has node set $V\left(G\right)\cup E\left(G\right)$, and lines joining all components of this node set which are contiguous or episode in G.

3 Main results

In this section we have discussed mainly about the central graph and total graph of central graph of generalized Petersen graph and their behaviors in harmonious coloring and matching number.

Theorem 3.1

For $k>2{\chi }_{h}\left\{C\left[GP\left(n,k\right)\right]\right\}=\left\{\begin{array}{c}\Delta \left\{C\left[GP\left(n,k\right)\right]\right\}+\Delta \left[GP\left(n,k\right)\right]+1,n\,is\,even\\ \Delta \left\{C\left[GP\left(n,k\right)\right]\right\}+\Delta \left[GP\left(n,k\right)\right]+2,n\,is\,odd\end{array}\right.$

Proof

Let $GP\left(n,k\right)$ be the Generalized Petersen graph with $2n$ nodes and $3n$ lines. Now each line of $GP\left(n,k\right)$ is now subdivided precisely once and the other non adjacent nodes are joined. Thus the number of nodes and lines of $C\left[GP\left(n,k\right)\right]$ are $5n$ and $2n\left(n+1\right)$ respectively.

Now we color these nodes in such a way that the maximum degree of $C\left[GP\left(n,k\right)\right]$ is $2n-1$ ie $4k+1$, hence we need $2n$ colors to colour these nodes, and when $n$ is even it is enough to color the left over $3n$ nodes with 3 colors and that is same as $\Delta \left[GP\left(n,k\right)\right]$, and when $n$ is odd it is enough to color the left over $3n$ nodes with 4 colors and that is same as $\Delta \left[GP\left(n,k\right)\right]+1$. Clearly we need $2n+3$ colors to color the central graph of Generalized Petersen graph $C\left[GP\left(n,k\right)\right]$, when $n$ is even and we need $2n+4$ colors to colour the central graph of Generalized Petersen graph $C\left[GP\left(n,k\right)\right]$, when $n$ is odd.

That is $\Delta \left\{C\left[GP\left(n,k\right)\right]\right\}+\Delta \left[GP\left(n,k\right)\right]+1$ colors when n is even and is $\Delta \left\{C\left[GP\left(n,k\right)\right]\right\}+\Delta \left[GP\left(n,k\right)\right]+2$ colors when n is odd.

Hence ${\chi }_{h}\left\{C\left[GP\left(n,k\right)\right]\right\}=\left\{\begin{array}{c}\Delta \left\{C\left[GP\left(n,k\right)\right]\right\}+\Delta \left[GP\left(n,k\right)\right]+1,n\,is\,even\\ \Delta \left\{C\left[GP\left(n,k\right)\right]\right\}+\Delta \left[GP\left(n,k\right)\right]+2,n\,is\,odd\end{array}\right.$

We manifest the result by taking induction on $k$.

For $k=3$ it is inevitable. Imagine the concept is true for $k=k$, then ${\chi }_{h}\left\{C\left[GP\left(n,k\right)\right]\right\}=\Delta \left\{C\left[GP\left(n,k\right)\right]\right\}+\Delta \left[GP\left(n,k\right)\right]+1$.

Let us confirm by using induction for $k=k+1$.

Consider $GP\left(n,k+1\right)$. A generalized Petersen graph with $n>2\left(k+1\right)$ that is $n=2k+3$. Hence number of nodes and lines in $GP\left(n,k+1\right)$ is increased by 4 and 6 respectively when compared to the nodes and lines of $GP\left(n,k\right)$. By $C\left(G\right)$, the 6 lines are subdivided by a new node. Removing these 10 nodes from $C\left[GP\left(n,k+1\right)\right]$, we get a subgraph G' which is nothing but $C\left[GP\left(n,k\right)\right]$. Hence by assumption we have $\chi_{h} \left\{ {C\left[ {G\prime } \right]} \right\} = \Delta \left\{ {C\left[ {G^{\prime}} \right]} \right\} + \Delta \left[ {G^{\prime}} \right] + 1$. Now by adding the removal nodes, $\Delta \{C\left[GP\left(n,k+1\right)\right]\}$ is $\Delta \left\{C\left[GP\left(n,k\right)\right]\right\}+4$ by the concept of central graph. Moreover $\Delta \left[GP\left(n,k\right)\right]$ is same as $\Delta \left[GP\left(n,k+1\right)\right]$.

Therefore

$${\chi }_{h}\left\{C\left[GP\left(n,k+1\right)\right]\right\}=\Delta \left\{C\left[GP\left(n,k\right)\right]\right\}+4+\Delta \left[GP\left(n,k+1\right)\right]+1$$

$$4k+1+4+\Delta \left[GP\left(n,k+1\right)\right]+1$$

$$4\left(k+1\right)+1+\Delta \left[GP\left(n,k+1\right)\right]+1$$

$$\Delta \left\{C\left[GP\left(n,k+1\right)\right]\right\}+\Delta \left[GP\left(n,k+1\right)\right]+1$$

Observation 3.2

For $k\ge 1$, $C\left[GP\left(n,k\right)\right]$ do not admit matching to be perfect and it has $n$ nodes which are not saturated.

$C\left[GP\left(\mathrm{4,1}\right)\right]$ with 4 unsaturated nodes Fig. 4.

Fig. 4

Illustration of Observation 3.2

Observation 3.3

Let $G$ be $C\left[GP\left(n,k\right)\right]$, then ${\alpha}^{^{\prime}}\left(G\right)=2n$.

Theorem 3.4

Let $G$ be $C\left[GP\left(n,k\right)\right]$, then for $k>2$ and $n$ even ${\chi }_{h}\left\{C\left[GP\left(n,k\right)\right]\right\}={\alpha}^{^{\prime}}\left(G\right)+\Delta \left[GP\left(n,k\right)\right]$ iff $G$ has $n$ unsaturated nodes, where $\alpha{}^{^{\prime}}$ is the maximum matching number.

Proof

Presume that ${\chi }_{h}\left\{C\left[GP\left(n,k\right)\right]\right\}$ is ${\alpha}^{^{\prime}}\left(G\right)+\Delta \left[GP\left(n,k\right)\right]$, then by the above result ${\alpha}^{^{\prime}}\left(G\right)=2n$ and $\Delta \left[GP\left(n,k\right)\right]=3$. Henceforth ${\alpha}^{^{\prime}}\left(G\right)=2n$ covers $2\left(2n\right)$ nodes, which is not up to $|G|$. Hence $G$ do not admit matching to be perfect. Hence there are $n$ nodes which are not saturated by 3.2.

Inversely, if $G$ has $n$ nodes which are not saturated then $G$ wont have a matching which is perfect and by the above result ${\alpha}^{^{\prime}}\left(G\right)=2n$. By 3.1 we have ${\chi }_{h}\left\{C\left[GP\left(n,k\right)\right]\right\}=\Delta \left\{C\left[GP\left(n,k\right)\right]\right\}+\Delta \left[GP\left(n,k\right)\right]+1$. Hence ${\chi }_{h}\left\{C\left[GP\left(n,k\right)\right]\right\}={\alpha}^{^{\prime}}\left(G\right)+\Delta \left[GP\left(n,k\right)\right]$.

Theorem 3.5

Let $G$ be $C\left[GP\left(n,k\right)\right]$, then for $k>2$ and $n$ odd ${\chi }_{h}\left\{C\left[GP\left(n,k\right)\right]\right\}={\alpha}^{^{\prime}}\left(G\right)+\Delta \left[GP\left(n,k\right)\right]+1$ iff $G$ has $n$ nodes which are not saturated, where ${\alpha}^{^{\prime}}$ is the maximum matching number.

Proof

Presume that ${\chi }_{h}\left\{C\left[GP\left(n,k\right)\right]\right\}$ is ${\alpha}^{^{\prime}}\left(G\right)+\Delta \left[GP\left(n,k\right)\right]+1$, then by the above result ${\alpha}^{^{\prime}}\left(G\right)=2n$ and $\Delta \left[GP\left(n,k\right)\right]=3$. Henceforth ${\alpha}^{^{\prime}}\left(G\right)=2n$ covers $2\left(2n\right)$ nodes, which is not up to $|G|$. Hence $G$ do not admit matching to be perfect. Hence there are $n$ nodes which are not saturated by 3.2.

Inversely, if $G$ has $n$ nodes which are not saturated then $G$ won’t have a matching which is not perfect and by the above result ${\alpha}^{^{\prime}}\left(G\right)=2n$. By theorem 3.1 we have ${\chi }_{h}\left\{C\left[GP\left(n,k\right)\right]\right\}=\Delta \left\{C\left[GP\left(n,k\right)\right]\right\}+\Delta \left[GP\left(n,k\right)\right]+2$. Hence ${\chi }_{h}\left\{C\left[GP\left(n,k\right)\right]\right\}={\alpha}^{^{\prime}}\left(G\right)+\Delta \left[GP\left(n,k\right)\right]+1$.

Corollary 3.6

Let $G$ be $C\left[GP\left(n,k\right)\right]$, then for $k>2$ and $n$ even, ${\chi }_{h}\left\{C\left[GP\left(n,k\right)\right]\right\}={\beta }^{^{\prime}}\left(G\right)+\Delta \left[GP\left(n,k\right)\right]-n$ iff $G$ has $n$ unsaturated nodes, where ${\beta }^{^{\prime}}$ is the minimum edge covering number.

Proof

We know that = ${\alpha}^{^{\prime}}+{\beta }^{^{\prime}}$ number of nodes. Thus from the above theorem it is clear that ${\chi }_{h}\left\{C\left[GP\left(n,k\right)\right]\right\}={\beta }^{^{\prime}}\left(G\right)+\Delta \left[GP\left(n,k\right)\right]-n$ iff $G$ has $n$ unsaturated nodes, where ${\beta }^{^{\prime}}$ is the minimum edge covering number.

Corollary 3.7

Let $G$ be $C\left[GP\left(n,k\right)\right]$, then for $k>2$ and $n$ odd, ${\chi }_{h}\left\{C\left[GP\left(n,k\right)\right]\right\}={\beta }^{^{\prime}}\left(G\right)+\Delta \left[GP\left(n,k\right)\right]-n+1$ iff $G$ has $n$ unsaturated nodes, where ${\beta }^{^{\prime}}$ is the minimum edge covering number.

Proof

We know that = ${\alpha}^{^{\prime}}+{\beta }^{^{\prime}}$ number of nodes. Thus from the above theorem it is clear that ${\chi }_{h}\left\{C\left[GP\left(n,k\right)\right]\right\}={\beta }^{^{\prime}}\left(G\right)+\Delta \left[GP\left(n,k\right)\right]-n+1$ iff $G$ has $n$ unsaturated nodes, where ${\beta }^{^{\prime}}$ is the minimum edge covering number.

Theorem 3.8

${\chi }_{h}\left\{T\left(C\left[GP\left(n,k\right)\right]\right)\right\}=\left\{\begin{array}{c}\frac{{\left(2n+1\right)}^{2}+2}{2}if\, n\, is\, odd\\ \frac{{\left(2n+1\right)}^{2}+3}{2}if\, n\, is\, even\end{array}\right.$

Proof

Let $\left\{{v}_{i}:i\in {Z}_{n}\right\}\cup \{{v}_{i}^{^{\prime}}:i\in {Z}_{n}\}$ be the nodes of $GP\left(n,k\right)$ so that $GP\left(n,k\right)$ has $2n$ nodes and $3n$ lines. Now each line of $GP\left(n,k\right)$ is now divided precisely once and the other non adjacent nodes are joined. Thus the number of nodes and lines of $C\left[GP\left(n,k\right)\right]$ are $5n$ and $2n\left(n+1\right)$ respectively.

Thus the vertex set of $C\left[GP\left(n,k\right)\right]$ is$\left\{ {v_{i} :i \in Z_{n} } \right\} \cup \left\{ {v_{i^\prime} :i \in Z_{n} } \right\} \cup \left\{ {u_{i} :i \in Z_{n} } \right\} \cup \left\{ {u_{i^\prime} :i \in Z_{n} } \right\} \cup \left\{ {u_{i^\prime \prime} :i \in Z_{n} } \right\}$ . The node ${u}_{i}$ is the division of the line ${v}_{i}{v}_{i+1}$ where $1\le i\le n-1$ and ${u}_{n}$ is a node of division of the line ${v}_{n}{v}_{1}$. Also let $v^\prime _{i} v^\prime _{{i + 1}} = u_{i} ^\prime \left( {1 \le i \le n - 1} \right)$ and ${u}_{n}^{^{\prime}}$ is a node of division of the line ${v}_{n}^{^{\prime}}{v}_{1}^{^{\prime}}$ and $v_{i} v^\prime_{i} = u_{i} ^{\prime\prime} (1 \le i \le n).$ Also let ${u}_{i}{v}_{i}={e}_{i}\left(1\le i\le n\right)$ and ${u}_{i}{v}_{i+1}={e}_{i}^{^{\prime}}\left(1\le i\le n-1\right)$ and ${u}_{n}{v}_{1}={e}_{n}^{^{\prime}}$ and ${u}_{i}^{^{\prime}}{v}_{i}^{^{\prime}}={f}_{i}\left(1\le i\le n\right)$ and ${u}_{i}^{^{\prime}}{v}_{i+1}^{^{\prime}}={f}_{i}^{^{\prime}}\left(1\le i\le n-1\right)$ and ${u}_{n}^{^{\prime}}{v}_{1}^{^{\prime}}={f}_{n}^{^{\prime}}$ and $v_{i} u_{i ^{\prime \prime}} = g_{i} \left( {1 \le i \le n} \right)$ and$u_{i} \prime \prime v_{i} \prime = g_{i} \prime \left( {1 \le i \le n} \right)$. Also let ${e}_{ij}={v}_{i}{v}_{j}$ and ${f}_{ij}={v}_{i}^{^{\prime}}{v}_{j}^{^{\prime}}$ and$g_{ij} = v_{i} v_{i^{\prime \prime}}$ . Thus $E\left\{C\left[GP\left(n,k\right)\right]\right\}=\left\{{e}_{i}:1\le i\le n\right\}\cup \left\{{e}_{i}^{^{\prime}}:1\le i\le n\right\}\cup {\{e}_{ij}:3\le i\le n-1\}\cup {\{e}_{ij}:2\le i\le n-2,i+2\le j\le n\}\cup \left\{{f}_{i}:1\le i\le n\right\}\cup \left\{{f}_{i}^{^{\prime}}:1\le i\le n\right\}\cup {\{f}_{ij}:3\le i\le n-1\}\cup {\{f}_{ij}:2\le i\le n-2,i+2\le j\le n\}\cup \left\{{g}_{i}:1\le i\le n\right\}\cup \left\{{g}_{i}^{^{\prime}}:1\le i\le n\right\}\cup {\{g}_{ij}:3\le i\le n-1\}\cup {\{g}_{ij}:2\le i\le n-2,i+2\le j\le n$.

By total graph the node set of $T\{C\left[GP\left(n,k\right)\right]\}$, is $2{n}^{2}+7n$ and lines joining all elements of this node set which are adjacent or incident in $C\left[GP\left(n,k\right)\right]$ is $n\left(4{n}^{2}+11\right)$. Hence the number of lines in $T\{C\left[GP\left(n,k\right)\right]\}$ is $n\left(4{n}^{2}+11n\right)$. Moreover in $T\{C\left[GP\left(n,k\right)\right]\}$, there are $3n$ nodes of degree 4, $6n$ nodes of degree $2n+1$ and $2n\left(n-1\right)$ nodes of degree $2\left(2n-1\right)$.

By total graph there exist a clique induced by the lines induced by ${v}_{i}$ in addition to them. Let ${K}_{2n}^{\left(i\right)}$ be the clique in $\{C\left[GP\left(n,k\right)\right]\}$). Each ${K}_{2n}^{\left(i\right)}$ shares exactly $2n\left(n-1\right)$ nodes with the remaining cliques. Thus harmonious coloring is performed with $2n$ distinct colors in ${K}_{2n}^{\left(1\right)}$. Since each clique shares some node with the other cliques the harmonious coloring in the remaining cliques are performed with $\left(2n-1\right),\left(2n-2\right)\dots \dots .$ colors. We manifest this by induction.

Case I

When $n$ is odd

Presume that the result is true when $n$ is odd. That is

${\chi }_{h}\left\{T\left(C\left[GP\left(n,k\right)\right]\right)\right\}=\frac{{\left(2n+1\right)}^{2}+2}{2}$. $GP\left(n+2,k\right)$ has new nodes ${v}_{n+1}$, ${v}_{n+2}$,${u}_{n+1}$ and ${u}_{n+2}$. These nodes together with the incident lines of ${v}_{n+1}$, ${v}_{n+2}$,${u}_{n+1}$ and ${u}_{n+2}$ forms four cliques of order $2n+2$ in $T\left(C\left[GP\left(n,k\right)\right]\right)$. The new vertices say ${v}_{n+1}$, ${v}_{n+2}$,${u}_{n+1}$,${u}_{n+2}$, ${e}_{i+1}$, $e_{{i + 2}} e_{{i + 1^\prime}}$, $e_{{i + 2^\prime}}$, ${f}_{i+1}$,${f}_{i+2}$, ${g}_{i+1}$, ${g}_{i+2}$, $g_{{i + 1^\prime}}$, $g_{{i + 2^{\prime\prime}}}$ are colored with 20 colors and the cliques ${K}_{2n}^{\left(i\right)}$, where i varies from 1 to 4 are colored with $8n-8$ colors. Therefore ${\chi }_{h}\left\{T\left(C\left[GP\left(n,k\right)\right]\right)\right\}=\frac{{\left(2n+1\right)}^{2}+2}{2}+8n-8+20=\frac{{\left(2n+1\right)}^{2}+2}{2}+8n+12$. Hence ${\chi }_{h}\left\{T\left(C\left[GP\left(n,k\right)\right]\right)\right\}=\frac{{\left(2n+5\right)}^{2}+2}{2}$.

Thus by induction we have

$${\chi }_{h}\left\{T\left(C\left[GP\left(n,k\right)\right]\right)\right\}=\frac{{\left(2n+1\right)}^{2}+2}{2}if\,n\,is\,odd$$

Case II

When $n$ is even

Presume that the result is true when $n$ is even. That is

${\chi }_{h}\left\{T\left(C\left[GP\left(n,k\right)\right]\right)\right\}=\frac{{\left(2n+1\right)}^{2}+3}{2}$. Now consider $GP\left(n+2,k\right)$ by acquainting four new nodes ${v}_{n+1}$, ${v}_{n+2}$,${u}_{n+1}$ and ${u}_{n+2}$. These nodes together with the incident lines of ${v}_{n+1}$, ${v}_{n+2}$,${u}_{n+1}$ and ${u}_{n+2}$ forms four cliques of order $2n+2$ in $T\left(C\left[GP\left(n,k\right)\right]\right)$. From central graph of total graph the new nodes say ${v}_{n+1}$, ${v}_{n+2}$,${u}_{n+1}$,${u}_{n+2}$, ${e}_{i+1}$, $e_{{i + 2}} e_{{i + 1}}^{\prime}$, $e_{{i + 2}} ^{\prime }$, ${f}_{i+1}$,${f}_{i+2}$, ${g}_{i+1}$, ${g}_{i+2}$, $g_{{i + 1}} ^{\prime }$, $g_{{i + 2^{\prime\prime}}}$ are assigned with 20 colors and the cliques ${K}_{2n}^{\left(i\right)}$, where $1\le i\le 4$ are assigned with $2n-2+2n-2+2n-2+2n-2=8n-8$ colors. Therefore ${\chi }_{h}\left\{T\left(C\left[GP\left(n,k\right)\right]\right)\right\}=\frac{{\left(2n+1\right)}^{2}+3}{2}+8n-8+20=\frac{{\left(2n+1\right)}^{2}+2}{2}+8n+12$. Hence ${\chi }_{h}\left\{T\left(C\left[GP\left(n,k\right)\right]\right)\right\}=\frac{{\left(2n+5\right)}^{2}+3}{2}$

Thus by induction we have

$${\chi }_{h}\left\{T\left(C\left[GP\left(n,k\right)\right]\right)\right\}=\frac{{\left(2n+1\right)}^{2}+3}{2},if\,n\,is\,even$$

4 Conclusion

In radio route framework which is one of the every now and again utilized aeronautics managing frameworks in awful climate conditions or on account of intangibility of ground objects. This framework depends on a system of extremely high recurrence Omni directional range radio reference points. To distinguish the present situation of a plane one needs to gauge the sign of two radio guides. Expect that state specialists chose to modernize the current system of radio signals. So as to lessen the expense of this venture they chose to introduce as hardly any kinds of signals as could reasonably be expected. Also every nation on the planet has formal systems of aviation route. Two contiguous hub of the system decide decisively one aviation route. By partner a node of graph G with each such hub and displaying every aviation route by a line of G, we acquire a chart relating to the system of aviation routes. Presently in the event that we discover ${\chi }_{h}$ of G, at that point the agreeable chromatic number will be actually the quantity of different kinds of guides mentioned by the state specialists. Henceforth the principle noteworthiness of our paper is that if the managing framework can be changed over to a diagram which appears as focal chart of snake determined systems then ${\chi }_{h}\left(G\right)$ gives the base number of radio reference points utilized for these controlling systems.

Therefore we have acquire the harmonious chromatic number of central graph of Generalized Petersen models. Additionally we have described the maximum matching number and mimum edge cover of central graph of Generalized Petersen models. Besides this work can be stretched out for understood models.