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On a Prime Labeling Conjecture.
Abstract
A graph with vertex set V is said to have a prime labeling if its vertices are labelled with distinct integers from {1,2......, |V|}such that for each edge xy, the labels assigned to x and y are relatively prime. A graph that admits a prime labeling is called a prime graph. It has been conjectured [1] that when n is a prime integer and m < n, the planar grid Pm × P n is prime. We prove the conjecture and also that P n×Pn is prime when n is a prime integer.
... Kanetkar [5] found prime labelings for the grid graph P n+1 ×P n+1 when n and (n + 1) 2 + 1 are primes, and either n = 5 or n ≡ 3 or 9 (mod 10), and a prime labeling on the grid graph P n × P n+2 when n ≡ 2 (mod 7) is prime. Sundaram et al. [7] have shown that the grid graph P p × P n has a prime labeling when p 5 is a prime and n p. They conjecture that the planar grid graph P m × P n is prime for all positive integers m and n. ...
... In this paper we show that, for any odd prime p and any positive integer p < n p 2 , the p × n planar grid graph P p × P n has a prime labeling. Combining this result with that of Sundaram et al. [7], for any odd prime p and any positive integer 1 n p 2 , the p × n planar grid graph P p × P n has a prime labeling. ...
... Sundaram et al. [7] have extended their result to the following theorem. ...
It is known that for any prime p and any integer n such that 1≤n≤p there exists a prime labeling on the pxn planar grid graph PpxPn. We show that PpxPn has a prime labeling for any odd prime p and any integer n such that that p is less than n≤p .
... Dean [7] and Ghorbani and Kamali [18] showed independently that all ladders are prime, resolving a conjecture of Varkey which was previously worked on in [38,36]. Other grid graphs have been shown to be prime, including P m P n if m ≤ n and n is prime [35], and a few other cases in [20]. It it true that P m P n is prime for all m and n? ...
... It it true that P m P n is prime for all m and n? This would settle a conjecture in [35]. Question 8.6. ...
A coprime labeling of a graph G is a labeling of the vertices of G with distinct integers from 1 to k such that adjacent vertices have coprime labels. The minimum coprime number of G is the least k for which such a labeling exists. In this paper, we determine the minimum coprime number for several well-studied classes of graphs, including the coronas of complete graphs with empty graphs, the joins of two paths, and prisms. In particular, we resolve a conjecture of Seoud, El Sonbaty, and Mahran and three conjectures of Asplund and Fox. We also provide bounds on the minimum coprime number of a random subgraph.
... One well-studied class of graphs within the field of prime labeling is the set of grid graphs, P m ×P n . It is conjectured that P m × P n is prime for all m, n ≥ 2, and many partial results have been proven for certain cases of m and n [12,18], and in particular ladder graphs, P m × P 2 , have been proven to be prime for any length m [5]. Ladder graphs were shown in [21] to be odd prime, but grid graphs with higher values of m have yet to be investigated for odd prime labelings. ...
An odd prime labeling is a variation of a prime labeling in which the vertices of a graph of order~n are labeled with the distinct odd integers 1 to so that the labels of adjacent vertices are relatively prime. This paper investigates many different classes of graphs including disjoint unions of cycles, stacked prisms, and particular types of caterpillars, by using various methods to construct odd prime labelings. We also demonstrate progress toward proving a conjecture that all prime graphs have an odd prime labeling.
... They call a graph G of order n is k -prime for some positive integer k if its vertices can labeled bijectively by the labels , 1,..., 1 k k k n such that adjacent vertices receive relatively prime labels. For known results on the prime labeling and its variations see [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. The reference [4] surveyed the known results to all variations of graph labelings appearing in this paper. ...
... The Prime labeling for planar grid is investigated by Sundaram et al [6], Lee.S.et.al [4] has proved that the wheel is a prime graph if and only if n is even. Duplication of a vertex of a graph produces a new graph by adding a vertex with ). ...
Graph theory is the one of the most important concept which takes the great roll in the electronic devices IC's. These components are called chips, contain complex, layered microcircuits that can be represented as sets of points interconnected by lines or arcs. Engineers develop integrated chips with maximum component density and minimum total interconnecting conductor length by using graph theory. This is important for optimizing processing speed and electrical efficiency in this paper advanced labeling of graph as implementation and also we have got new results on Fibonacci graceful labeling graphs which helps in networking and electronic circuitry a lot.
... Roger surmised during the period 1980s that all trees possess prime labeling, and what he surmised could not be confirmed as a fact till now. Sundaram [8] is one of the exponents who studied the prime labeling for planner grid. Further investigations included the development of prime labelings by authors such as Ganesan and Balamurugan [4] who developed prime labellings for Theta graphs and Meena and Vaithilingam [6] for graphs related to Helm. ...
The paper investigates prime labeling of Jahangir graph J n,m for n ≥ 2, m ≥ 3 provided that nm is even. We discuss prime labeling of some graph operations viz. Fusion, Switching and Duplication to prove that the Fusion of two vertices v 1 and v k where k is odd in a Ja-hangir graph J n,m results to prime graph provided that the product nm is even and is relatively prime to k. The Fusion of two vertices v nm + 1 and v k for any k in J n, m is prime. The switching of v k in the cycle C nm of the Jahangir graph J n,m is a prime graph provided that nm+1 is a prime number and the switching of v nm+1 in J n, m is also a prime graph .Duplicating of v k, where k is odd integer and nm + 2 is relatively prime to k,k+2 in J n,m is a prime graph.
... Roger surmised during the period 1980s that all trees possess prime labeling, and what he surmised could not be confirmed as a fact till now. Sundaram [8] is one of the exponents who studied the prime labeling for planner grid. Further investigations included the development of prime labelings by authors such as Ganesan and Balamurugan [4] who developed prime labellings for Theta graphs and Meena and Vaithilingam [6] for graphs related to Helm. ...
The paper investigates prime labeling of Jahangir graph Jn,m for n ≥ 2, m ≥ 3 provided that nm is even. We discuss prime labeling of some graph operations viz. Fusion, Switching and Duplication to prove that the Fusion of two vertices v1 and vk where k is odd in a Jahangir graph Jn,m results to prime graph provided that the product nm is even and is relatively prime to k. The Fusion of two vertices vnm + 1 and vk for any k in Jn, m is prime. The switching of vk in the cycle Cnm of the Jahangir graph Jn,m is a prime graph provided that nm+1 is a prime number and the switching of vnm+1 in Jn, m is also a prime graph .Duplicating of vk, where k is odd integer and nm + 2 is relatively prime to k,k+2 in Jn,m is a prime graph.
... M. Sundaram et.al. [13] Proved that when n is a prime integer and , the planar grid is prime and also proved is prime, when n is a prime integer [5]. Alka V Kanetkar [1] proved that the grid has a prime labeling, when n is an odd prime, and is also a prime. ...
... We notice that in the survey [12] it is incorrectly stated that this had been proved in [19]. Partial results on this conjecture appeared in [4,15,16,19]. Our goal is to prove the conjecture. ...
A prime labeling of a graph with n vertices is a labeling of its vertices with distinct integers from in such a way that the labels of any two adjacent vertices are relatively prime. T. Varkey conjectured that ladder graphs have prime labeling. We prove this conjecture.
In this paper, we introduce new results on k-prime labeling. First, we discuss the kprime labeling of cycles Cn for some values of k and n . Also we give the k-prime labeling of combs Pn K1and some case of the crowns Cn K1 . Second, we show that all wheels W 2n> 1 are not k-prime for every positive integers k and n while W2n (n 2) is not k-prime for every even positive integers k . Finally, we give the k-prime labeling of the helm Hn (n 3) for 2 k 11 and we show that if k 2(n 1) and k 2n are twin primes where n 3 and k 1, then Hn is k -prime.
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