The odd harmonious labeling of dumbbell and generalized prism graphs

Gusti A. Saputri, Kiki A. Sugeng, Dalibor Froncek

Research output: Contribution to journalArticlepeer-review

10 Scopus citations

Abstract

A graph G = (V, E) with {pipe}E{pipe} = q is said to be odd harmonious if there exists 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* (xy) = f(x) + f(y) is a bijection. Then f is said to be odd harmonious labeling of G. A dumbbell graph Dn,k,2 is a bicyclic graph consisting of two vertex-disjoint cycles Cn, Ck and a path P2 joining one vertex of Cn with one vertex of Ck. A prism graph Cn × Pm is a Cartesian product of cycle Cn and path Pm. In this paper we show that the dumbbell graph Dn,k,2 is odd harmonious for n ≡ k ≡ 0 (mod 4) and n ≡ k ≡ 2 (mod 4), generalized prism graph Cn × Pm is odd harmonious for n ≡ 0 (mod 4) and for any m, and generalized prism graph Cn × Pm is not odd harmonious for n ≡ 2 (mod 4).

Original languageEnglish (US)
Pages (from-to)221-228
Number of pages8
JournalAKCE International Journal of Graphs and Combinatorics
Volume10
Issue number2
StatePublished - Aug 1 2013

Keywords

  • Dumbbell
  • Generalized prism
  • Odd harmonious graphs
  • Odd harmonious labeling

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