Prime labeling of line and splitting graph of brush graph

A bijective function f from V(G) to {1,2,…, n} be a prime labeling of a graph G with n order if for every u, v ∈ V(G) such that e = uv ∈ E(G), f(u) and f(v) relatively prime. A prime graph is a graph which admits prime labeling. In this study, we investigate and conclude that the line and splitting graph of the brush graph is a prime graph.

NEW RESULTS ON K -PRIME LABELING

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.

On SD-Prime Labelings of Some One Point Unions of Gears

Let G=(V(G),E(G)) be a simple, finite and undirected graph of order n. Given a bijection f:V(G)→{1,…,n}, and every edge uv in E(G), let S=f(u)+f(v) and D=|f(u)−f(v)|. The labeling f induces an edge labeling f′:E(G)→{0,1} such that for an edge uv in E(G), f′(uv)=1 if gcd(S,D)=1, and f′(uv)=0 otherwise. Such a labeling is called an SD-prime labeling if f′(uv)=1 for all uv∈E(G). We provide SD-prime labelings for some one point unions of gear graphs.

On a Question of Prime Labeling of Graphs

In the field of graph theory, the complete graph of vertices is a simple undirected graph such that every pair of distinct vertices is connected by a unique edge. In the present work, we introduce planar subgraph of with maximal number of edges . Generally, does not admit prime labeling. We present an algorithm to obtain prime-labeled subgraphs of . We conclude the paper by stating two conjectures based on labeling of . In particular, the planar subgraph admits anti-magic labeling but does not admit edge magic total labeling.