On Tensor Product and Colorability of Graphs

The idea of graph coloring problem (GCP) plays a vital role in allotment of resources resulting in its proper utilization in saving labor, space, time and cost effective, etc. The concept of GCP for graph G is assigning minimum number of colors to its nodes such that adjacent nodes are allotted a different color, the smallest of which is known as its chromatic number χ ( G ) . This work considers the approach of taking the tensor product between two graphs which emerges as a complex graph and it drives the idea of dealing with complexity. The load balancing on such complex networks is a hefty task. Amidst the various methods in graph theory the coloring is a quite simpler tool to unveil the intricate challenging networks. Further the node coloring helps in classifying the nodes with least number of classes in any network. So coloring is applied to balance the allocations in such complex network. We construct the tensor product between two graphs like path with wheel and helm, cycle with sunlet and closed helm graphs then structured their nature. The coloring is then applied for the nodes of the extended new graph to determine their optimal bounds. Hence we obtain the chromatic number for the tensor product of P m ⊗ W n , P m ⊗ H n , C m ⊗ S n and C m ⊗ CH n .


Introduction
A graph represents a set of objects called nodes or vertices in which some pairs of nodes are connected by edges. GCP is one of the earlier and interesting concepts in graph theory initially started with coloring the regions of every map. Many researches were done for the past few decades on various coloring concepts such as vertex coloring, edge coloring, total coloring, equitable coloring and star coloring, etc. It has been investigated in the form of optimal assignment by finding the chromatic number and developing coloring algorithms. Murat et al. [6] studied the performance of various graph coloring algorithms based on their solution and execution time. Graph coloring is widely applied in the process of frequency assignment, task scheduling and load balancing etc. Jong et al. [4] designed and implemented a load balancing approach in edge cloud computing environments with GCP based on a genetic algorithm.
Very few research works has been carried out in the graph theoretical concepts like product of graphs and node coloring due to its complexity. The tensor product (TP) of graphs with node coloring is a recent approach. We considered the combination of TP and its chromatic number of graphs. For our study we considered the TP of path with wheel and helm graphs, cycle with sunlet and closed helm graph and applied node coloring to obtain χ(G). The coloring is done by the method of mapping the node set V in the TP of two different graphs to the color set C . This mapping transforms into a proper node coloring G. Reed [8] established the relation among the bounds of chromatic number χ, maximum degree ∆ and maximum clique number ω of any graph G. Further this Reed's conjecture is studied for the TP of graphs P m ⊗ W n , P m ⊗ H n , C m ⊗ S n and C m ⊗ CH n . These minimum bounds will resolve the allocation problems in larger and complex networks and enhance the performance with minimum resources.
The Wheel graph n ≥ 4 is obtained by connecting all the nodes {v 1 , v 2 , . . . , v n } of the cycle C n to the hub node v 0 .
The Helm graph [3] H n is acquired from a Wheel graph W n in which each node on the outer cycle C n is joined with a pendant edge.
A Closed Helm [3] CH n is obtained from a helm H n after joining each pendant node with an edge between them.
The Sunlet graph S n [10] is the graph with 2n nodes con-sisting of a cycle C n and n pendant edges each connected to a node {v 1 , v 2 , . . . , v n } of the cycle. The Proper Node Coloring [5] of a graph G (V, E) is defined as i.e., no two neighboring nodes receives the identical color. The least number of colors to which the proper node coloring exists for a graph G is known as the chromatic number χ(G).
The Tensor product(TP) [11] of graphs G and H is represented as G ⊗ H, whose elements are where each node (g, h) and (g ′ , h ′ ) are adjacent precisely if gg ′ ∈ E (G) and hh ′ ∈ E (H).
Theorem [2] [Brooks Theorem] Any graph G which is connected, apart from a complete graph K n or an odd cycle C n , n ≡ 1 mod 2 then χ (G) ≤ ∆ (G).
Theorem [9] For any two graphs G 1 and G 2 which are connected, then its TP G 1 ⊗ G 2 is also a connected graph if and only if either G 1 or G 2 consists of an odd cycle.
Corollary [9] For two graphs G 1 and G 2 which are connected without having any odd cycles, then its TP G 1 ⊗ G 2 has absolutely two connected components.
2 Node coloring for tensor product of graphs Theorem 1 For any positive integer values of m ≥ 2 and n ≥ 3, the chromatic number for TP of path P m and wheel graph W n is χ(P m ⊗ W n ) = 2.
Proof The path P m has m nodes and m − 1 edges and is represented as The wheel W n contains of n + 1 nodes and 2n edges which are defined as . By considering the TP of these two graphs, it generates a product graph with m(n + 1) nodes and 4(m − 1)n edges such that The construction of TP with P m , m ≥ 2 and wheel graph W n , n ≥ 3 is shown in Figure 1.
The node coloring are made as follows From the above process of coloring as shown in Figure 1, the generalized structure of TP between path P m , m ≥ 2 and wheel graph W n , n ≥ 3. It is observed that it requires only 2 colors to color the entire graph. Therefore the chromatic number of this tensor product of path and wheel graphs is 2. Hence χ(P m ⊗ W n ) = 2.
Remark 1 For any m ≥ 2, n ≥ 3 the tensor product between the two graphs path P m and W n with the maximum degree ∆ and the clique number ω the Reed's conjecture χ(P m ⊗ W n ) ≤ ⌈ 1+∆+ω 2 ⌉ holds. For P m ⊗ W n (m ≥ 2, n ≥ 3), it is observed that ∆ = 6 and ω = 2 which satisfies the Reed's conjecture χ(P m ⊗ W n ) ≤ 5.
Theorem 2 For any positive integer m ≥ 2 and n ≥ 3, the chromatic number for TP of path P m and helm graph H n is χ(P m ⊗ H n ) = 2.
Proof The elements of the path is formulated as The helm graph H n contains 2n + 1 nodes and 3n edges which is defined as where the edges e i are between the nodes v 0 v i (1 ≤ i ≤ n), the edge e ′ i is between the nodes v i v i+1 (1 ≤ i ≤ n − 1) and the edges e ′′ i are between the nodes v i v n+i (1 ≤ i ≤ n). By considering the TP of these two graphs, it results in a bigger graph with m(2n + 1) nodes and 6(m − 1)n edges such that From this method of coloring the TP as graphed in Figure  2, is easy to state that it requires only 2 colors for coloring the entire graph of any order m ≥ 2 and n ≥ 3. Therefore χ(P m ⊗ H n ) = 2. Remark 2 For the tensor product between path P m and H n ,(m ≥ 2, n ≥ 3) the maximum degree ∆ and the clique number ω is observed that ∆ = 8 and ω = 2 which satisfies the Reed's conjecture χ(P m ⊗ W n ) ≤ 6.
Theorem 3 For any positive integers of m, n ≥ 3, Proof The cycle C m consists both nodes and edges of order and size m which is represented as The sunlet graph S n consists both nodes and edges of order and size 2n which is formulated as , the edge e n is between v n v 1 and the edges e ′ i are between the nodes v i v n+i (1 ≤ i ≤ n − 1). Taking TP for the above two graphs, the resultant graph has 2mn nodes and 4mn edges, therefore On Tensor Product and Colorability of Graphs The node coloring is processed by defining f : P → C , The coloring on the TP of odd cycle C m and sunlet S n m, n ≥ 3 is diagrammed in Figure 3. Further the coloring on the generalized structure of tensor product with even cycle C m , m > 3 and sunlet S n , n ≥ 3 is portrayed in Figure 4. Hence it is observed that, while taking tensor product with cycles it requires 3 colors for odd cycles and 2 colors for even cycles.
Therefore χ (C m ⊗ S n ) = 3, if m is odd 2, if m is even .

Remark 3
The tensor product between the cycle C m and Sunlet graph S n ,(m, n ≥ 3) has the chromatic number 3 if m = odd and 2 if n = even. The maximum degree is ∆ = 6 and the clique number ω = 2. It is evident that Reed's conjecture holds for this product graph in both odd and even cases, which implies χ(C m ⊗ S n ) ≤ 5. Proof The cycle is formulated as The closed helm CH n has 2n + 1 nodes and 4n edges and defined as where the edges e i are between the nodes v 0 v i (1 ≤ i ≤ n), similarly the edges e ′ i are between the nodes v i v i+1 (1 ≤ i ≤ n − 1), the edge e ′ k is between the nodes v n v 1 , the edges e ′′ i are between the nodes v i v n+i (1 ≤ i ≤ n), the edges e ′′′ i are between the nodes v i v i+1 (n + 1 ≤ i ≤ 2n − 1) and the edge e ′′′ k is between the nodes v 2n v n+1 .
By taking the tensor product for these two graphs, the extended new graph with m(2n+1) nodes and 8mn edges which has The node coloring is obtained by considering a function f : The coloring on the TP of odd cycle C m and closed helm CH n m, n ≥ 3 is presented in Figure 5. The summarizing of the coloring on the generalized construction of tensor product with even cycle C m , m > 3 and sunlet CH n , n ≥ 3 is exhibited in Figure 6. Hence it is clear that while taking TP of cycles with closed helm graph, it requires 3 colors for odd cycles and 2 colors for even cycles.

Remark 4
The TP between the cycle C m and closed helm CH n ,(m, n ≥ 3) has the chromatic number 3 if m is odd and 2 if m is even. The maximum degree is ∆ = 8 and the clique number ω = 3. It is clear that Reed's conjecture is true for this graph χ(C m ⊗ S n ) ≤ 6.

Conclusions
In this paper we have applied vertex coloring to obtain the chromatic number of TP of graphs pondered. We have shown the minimum colors required for optimal allocation to the tensor product between different graphs like path, cycle, helm, closed helm and sunlet graph. Furthermore bounds of chromatic number could be explored and achieved for tensor products between different graphs.