**Journal of Prime Research in Mathematics**

Vol. 1 (2010), Issue 1, pp. 01 – 12

ISSN: 1817-3462 (Online) 1818-5495 (Print)

ISSN: 1817-3462 (Online) 1818-5495 (Print)

# Divisor path decomposition number of a graph

**K. Nagarajan
**Department of Mathematics, Sri S.R.N.M.College, Sattur – 626 203, Tamil Nadu, India.

**Department of Mathematics, V.O.C.College, Tuticorin – 628 008, Tamil Nadu, India.**

A. Nagarajan

A. Nagarajan

\(^{1}\)Corresponding Author: k_nagarajan_srnmc@yahoo.co.in

Copyright © 2010 K. Nagarajan, A. Nagarajan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Published:**December, 2010.

### Abstract

A decomposition of a graph G is a collection Ψ of edge-disjoint subgraphs \(H_1,H_2, . . . , H_n\) of \(G\) such that every edge of \(G\) belongs to exactly one \(H_i\). If each \(H_i\) is a path in \(G\), then \(Ψ\) is called a path partition or path cover or path decomposition of \(G\). A divisor path decomposition of a \((p, q)\) graph \(G\) is a path cover \(Ψ\) of \(G\) such that the length of all the paths in \(Ψ\) divides \(q\). The minimum cardinality of a divisor path decomposition of \(G\) is called the divisor path decomposition number of \(G\) and is denoted by \(π_D(G)\). In this paper, we initiate a study of the parameter \(π_D\) and determine the value of \(π_D\) for some standard graphs. Further, we obtain some bounds for \(π_D\) and characterize graphs attaining the bounds.

#### Keywords:

Divisor path, greatest divisor path, divisor path decomposition, divisor path decomposition number.