(1,N)- Arithmetic Labelling of n-polygonal snakes and P(a,b)

Abstract

A (p,q) - graph G is said to have (1, N) - Arithmetic labelling if there is a function φ from the vertex set V (G) to

{0, 1, N,(N + 1), 2N,(2N + 1), ...,(q q 1) N,(q q 1) N + 1} so that the values of the edges, obtained as the sums of the labelling assigned to their end verices can be arranged in the arithmetic progression

1,(N + 1),(2N + 1), ...,(q q 1) N + 1.  In this paper we prove that the n-polygonal snakes for n ≡ 0 (mod4) and

P (a, b) for even values of a and odd values of b have (1, N) ) arithmetic labelling for every positive integer N > 1.