(1, N)-Arithmetic Labelling of Arbitrary Supersubdivision of disconnected graphs

Graph Theory

Abstract

A (p, q) -graph G is said to be (1, N) -Arithmetic if there is a function φ from the vertex set V (G) to {0, 1, N,(N + 1), 2N,(2N + 1), . . . , N(q -1), N(q-1) + 1}

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

1) + 1} . In this paper we prove that the arbitrary supersubdivision of disconnected paths Pn ∪Pr and disconnected path and cycle Pn∪C r  are (1, N) -Arithmetic Labelling for all positive integers N > 1.