Gaussian Neighborhood-prime Labeling of Graphs Containing Hamiltonian Cycle

In this paper, we examine Gaussian neighborhood-prime labeling of generalized Peterson graph and graphs which contain Hamiltonian cycle.


Introduction
The extension of prime labeling on natural numbers to the set of Gaussian integer is known as Gaussian prime labeling. The concept of Gaussian prime labeling with the help of spiral ordering of Gaussian integer was firstly introduced by Hunter Lehmann and Andrew Park in . In this paper, Hunter Lehmann and Andrew Park gave a milestone result that any tree with ≤72 vertices is Gaussian prime tree under the spiral order. Steven Klee, Hunter Lehmann and Andrew Park (Klee, Lehmann and Park 2016) also proved that the path graph, star graph, n-centipede tree, (n, m, k) double star tree, (n, 3) firecracker tree are Gaussian prime graphs. We all know the Entringer (Robertson and Small 2009) conjecture that any tree admits prime labeling, but this conjecture has not yet been proven for all trees. S.K. Patel and N. P. Shrimali (Patel and Shrimali 2015) introduced one of the variation of prime labeling, which is neighborhood-prime labeling of a graph. They proved that following graphs are neighborhood-prime: path, complete, wheels, Helms, closed Helm, flowers, certain union of cycles. (Patel 2017)proved that Generalized Petersen graphs are neighborhood-prime graphs for certain cases. Malori Cloys and N. Bradley Fox (Cloys and Fox 2018) almost covered large class of trees which have neighborhood-prime labeling such as caterpillars, spiders, firecrackers and any tree that contains no two degree vertices.
In addition, Malori Cloys and N. Bradley Fox put forth conjecture that all trees are neighborhood-prime. Similar conjecture was made by Entriger for prime labelings. John Asplund, N. Bradley Fox and Arran Hamm (Asplund and Fox 2018) well-built the result that any graph containing Hamiltonian cycle is neighborhood-prime graph. With the help of Hamiltonicity, they proved that the Generalized Petersen graph GP (n, k) is neighborhood-prime graph for all n and k. The detailed list of neighborhood-prime graph is also available in the dynamic survey of graph labeling written by (Gallian 2016).
The Gaussian neighborhood-prime labeling was firstly introduced by (Rajesh Kumar and Mathew Varkey 2018) with respect to spiral order. Rajesh Kumar et al. initiated their work by showing the graphs like: path, star, (p, n, m) double star tree with n ≤ m, comb P n  K 1 , Spiders, (n, 2) centipede tree, cycles C n with n ≡ 2 4 ( ) mod are Gaussian neighborhood-prime graph under spiral ordering of Gaussian integers. In this paper, we investigate Gaussian neighborhood-prime labeling of graph containing Hamiltonian cycle. We will discuss more results depending upon the Hamiltonicity which guarantees that the graph is Gaussian neighborhood-prime graph under the spiral order. Further, we will prove that generalized Petersen graphs are Gaussian neighborhood-prime graphs under the spiral order. pp.162 We begin with some definitions and the background of Gaussian integer before introducing main results. We will use spiral ordering of Gaussian integers and its properties given by (Steven Klee et al. 2016).

Background of Gaussian Integer and Spiral Ordering
The complex numbers of the form γ = + ∈ p iq; p, q  are known as Gaussian integer. We denote set of Gaussian (Steven Klee et al. 2016) introduced the Spiral ordering of the Gaussian integer and defined g n+1 recursively starting with g 1 = 1 as follows: where g n denote the n th Gaussian integer with above ordering.
In notation, we write first 'n' Gaussian integers by [g n ].
In (Steven Klee et al. 2016) had already established some useful properties about Gaussian integers with the above ordering like : • Any two consecutive integers are relatively prime.
• Any two consecutive odd integers are relatively prime. • g and γ µ + + ( ) 1 i k are relatively prime, if g is an odd Gaussian integer and m is a unit. where k is a positive integer.
Definition 2.1: (Gross and Yellen 1999) The set of all vertices in G which are adjacent to u is called neighborhood of vertex u. In notation, we write N(u). ( )} g u u N w ∈ are relatively prime for every vertices w V G ∈ ( ) with degree greater than one. A graph which admits Gaussian neighborhoodprime labeling is known as Gaussian neighborhood-prime graph.

Definition 2.3 : (Cloys and Fox 2018)
The size of largest cycle in graph G is called circumference of a graph G.
In this paper, we considered all graphs which are undirected, finite and simple. For the notations and terminology of graph theory, we have referred (Gross and Yellen 1999). Throughout this paper, we will understand that the graph is a Gaussian neighborhood-prime graph meant to be a Gaussian neighborhood-prime graph with the spiral ordering.

Main Results
Theorem 3.1 If H is a Hamiltonian graph having n vertices with n ≡ 2 ( 4) mod , then H is a Gaussian neighborhoodprime graph.
Proof: Firstly, we note that if H is a Gaussian neighborhoodprime graph then the new graph formed by adding an edge in H between two vertices of degree at least two is again Gaussian neighborhood-prime graph.
In order to prove H has Gaussian neighborhoodprime labeling it is enough to show that the Hamiltonian cycle of H has Gaussian neighborhood-prime labeling. Let C v v n = ( , ...., ) 1 v 2 be the Hamiltonian cycle in graph H. Now, we reformulate the labeling of cycles used by (Rajesh Kumar and Mathew Varkey 2018) in as follows: Define Note that each vertex v i i ( ) ≠ 1 of C there exist two neighbors of v i whose labels are consecutive Gaussian integers. The neighbors of v 1 are v 2 , v n , having labels g 1 , g n respectively. Hence, Hamiltonian cycle C has Gaussian neighborhoodprime labeling if n is not congruent to 2 modulo 4. Which completes the proof. pp.163 Theorem 3.2 Let H be a Gaussian neighborhood-prime graph with n vertices, where n is not congruent to 2 modulo 4, having Hamiltonian cycle C = (u 1 , u 2 .... u n ). If graph H is obtained from H using additional k vertices {w 1 , w 2 ,...., w k } in such a way that each w j is adjacent to u m i and u m i +2 where subscripts of V(C) are calculated under modulo n then H is Gaussian neighborhood prime graph. ∈ are relatively prime. As each w j is adjacent to u m i and u m i+2 which are either labeled by consecutive Gaussian integers or one of them contains the label g 1 . Thus, h is a Gaussian neighborhood-prime labeling. Consequently, H is a Gaussian neighborhood-prime graph.

Theorem 3.3 If H is a Hamiltonian graph having an odd cycle then H has Gaussian neighborhood-prime labeling.
Proof: Let H be a Hamiltonian graph having n vertices. forms an odd cycle. We assign the labels to vertices u 1 , u 3 , u 5 ,..., u n-1 by γ γ γ Theorem 3.4 If G is a connected graph with n vertices such that n ≡ 3 ( 4) mod and G has circumference n -1 then G has a Gaussian neighborhood-prime labeling.
Proof: Let C = (w 1 , w 2 ..... w n-1 ) be the cycle and w be the vertex in graph G which does not lie on cycle C. We have following cases for vertex w: Case-I: If deg(w) = 1 then we define the labeling (In which n is replaced by n -1) and h(w) = g n . The reader can easily verify that h is a Gaussian neighborhoodprime labeling. Case-II: If deg(w) > 1 then without loss of generality we assume that w is adjacent to w k (1 ≤ k ≤ n -1) on cycle C. Define a bijection h: and h(w) = g n . where the subscript i + (k -2) is calculated under modulo n -1. From Equation (3), one can see that The other vertices of cycle C consists two neighbors whose labels are consecutive Gaussian integers. In both the cases, we have cycle if length n -1 which is Gaussian neighborhoodprime graph if n − ≡ 1 2 4 ( ) mod . Thus, h admits Gaussian neighborhood-prime labeling if n ≡ 3 ( 4) mod .
(Cloy and Fox 2018) proved GP n n , 2 neighborhood-prime graph. We use the idea of the neighborhood-prime labeling of GP n n , 2 For each 0 ≤ i < n -1, the vertex u i has neighbors v i and u i+1 which are also labeled by consecutive Gaussian integers. The vertices u n--1 has neighbor u 0 whose label is g 1 . Thus, relatively prime Gaussian integers. Therefore, h is a Gaussian neighborhood-prime labeling. (Patel 2017) proved GP(n, k) is neighborhood-prime graph if n and k are relatively prime, we will use the approach of that labeling in the following lemma.
Let w be any vertex of V(GP(n, k)), we claim that the Gaussian integers in the set {h(x): x ∈ N(w)} are relatively prime.

Case-
The neighbors of exterior vertices v j (j ≠ 2, n) are v j-1 , v j+1 having labels h(v j-1 ) and h(v j+1 ) which are consecutive odd Gaussian integers. The vertices v 2 and v n have common neighbor v 1 whose label is g 1 . Case-2: w = u j , 1 ≤ j ≤ n The neighbors of internal vertices u j are v j , u j+k , u j+(k+1) where the subscripts of vertices u j are reduced modulo n. From the Equations (4), (5), (6), (7) observe that for each u j (j ≠ 1), either h(v j ) and h(u j+k ) is consecutive Gaussian integers or h(v j ) and h(u j+k+1 ) are consecutive Gaussian integers. Finally, one of the neighbor of vertex u 1 is v 1 with label g 1 .
Thus, the set {h(x): x ∈ N(w)} consists relatively prime Gaussian integers in each of the above cases which implies that h is pp.165 a Gaussian neighborhood prime labeling. Figure 1. Gaussian Neighborhood-prime labeling of the GP(11.5) and GP(17.8) Theorem 3.7 For each n and k, the generalized Peterson graph GP(n, k) has Gausssian neighborhood-prime labeling.
Proof: In (Alspach 1983) proved that for each n and k, the graph GP(n, k) is Hamiltonian except following two cases: • n ≡ 0 (mod 4) and n ≥ 8 with k n = 2 .
The proof follows from Theorem 3.1 and Theorem 3.3 together with previous two lemmas.