Direct product primality testing of graphs is GI-hard

On (Semi-) Edge-primality of Graphs

Let $G= (V,E)$ be a $(p,q)$-graph. A bijection $f: Eto{1,2,3,ldots,q }$ is called an edge-prime labeling if for each edge $uv$ in $E$, we have $GCD(f^+(u),f^+(v))=1$ where $f^+(u) = sum_{uwin E} f(uw)$. Moreover, a bijection $f: Eto{1,2,3,ldots,q }$ is called a semi-edge-prime labeling if for each edge $uv$ in $E$, we have $GCD(f^+(u),f^+(v))=1$ or $f^+(u)=f^+(v)$. A graph that admits an  ...

Primality testing

Primality Testing

Prime numbers are extremely useful, and are an essential input to many algorithms in large part due to the algebraic structure of arithmetic modulo a prime. In everyday life, perhaps the most frequent use for prime numbers is in RSA encryption, which requires quite large primes (typically≥ 128-bits long). Fortunately, there are lots of primes—for large n, the probability that a random integer l...

Independence in Direct-Product Graphs

Let α(G) denote the independence number of a graph G and let G ×H be the direct product of graphs G and H. Set α(G ×H) = max{α(G) · |H|, α(H) · |G|}. If G is a path or a cycle and H is a path or a cycle then α(G×H) = α(G×H). Moreover, this equality holds also in the case when G is a bipartite graph with a perfect matching and H is a traceable graph. However, for any graph G with at least one ed...

Algorithmic Primality Testing

This project consists of the implementation and analysis of a number of primality testing algorithms from Chapter 9 of 3]. The rst algorithm is added to provide a point of comparison and the section on Lucas-Lehmer testing was derived mainly from material found in 4]. Aside from simply implementing these tests in Maple and C, the goal of this paper is to provide a somewhat simpliied and more ge...