Title: On odd harmonious labelling of even cycles with parallel chords and dragons with parallel chords

Authors: V. Srividya; R. Govindarajan

Addresses: Department of Mathematics, Bharathi Women's College, Chennai, 600-108, India ' Department of Mathematics, D.G. Vaishnav College, Chennai, 600-106, India

Abstract: Labelling in graph theory is an active area of research due to its wide range of applications. A graph labelling is an assignment of integers to the vertices (or) edges (or) both subject to certain conditions. This paper deals with one such labelling called odd harmonious labelling. A graph G = (V, E) with |V (G)| = p and |E (G)| = q is said to be odd harmonious if there exist an injection f : V (G) → {0, 1, 2, …, 2q – 1} such that the induced function f* : E (G) → {1, 3, 5, …, 2q – 1} defined by f*(uv) = f(u) + f(v) is bijective. In this paper we prove that every even cycle C_n (n ≥ 6) with parallel P₃ chords is odd harmonious. We also prove that the disjoint union of two copies of even cycle C_n with parallel P₃ chords and the joint sum of two copies of even cycle C_n with parallel P₃ chords is odd harmonious. Moreover we show that the chain of even cycles C_n (n ≥ 6) with parallel P₃ chords, joining two copies of even cycles C_n by a path and also dragons with parallel chords obtained from every odd cycle C_n (n ≥ 7) after removing two edges from the cycle C_n, dragons with parallel P₄ chords obtained from every odd cycle C_n (n ≥ 9) after removing two edges from the cycle C_n are odd harmonious.

Keywords: harmonious labelling; odd harmonious labelling; joint sum; chain of cycles; dragons.

DOI: 10.1504/IJCAET.2020.110475

International Journal of Computer Aided Engineering and Technology, 2020 Vol.13 No.4, pp.409 - 424

Accepted: 23 Jan 2018
Published online: 22 Oct 2020 *

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