Journal of Discrete Mathematics

Volume 2013 (2013), Article ID 721051, 6 pages

http://dx.doi.org/10.1155/2013/721051

## Decomposition of Graphs into Paths and Cycles

^{1}Core Group Research Facility (CGRF), National Center for Advanced Research in Discrete Mathematics (n-CARDMATH), Kalasalingam University, Anand Nagar, Krishnankoil 626190, India^{2}Department of Mathematics, The Madura College, Madurai 625 011, India^{3}Department of Mathematics, Christ University, Bangalore 560 029, India

Received 31 October 2012; Revised 12 March 2013; Accepted 12 March 2013

Academic Editor: Kinkar Ch Das

Copyright © 2013 S. Arumugam et al. 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.

#### Abstract

A *decomposition* of a graph is a collection of edge-disjoint subgraphs of such that every edge of belongs to exactly one . If each is a path or a cycle in , then is called a *path decomposition* of . If each is a path in , then is called an *acyclic path decomposition* of . The minimum cardinality of a *path decomposition* (acyclic path decomposition) of is called the *path decomposition number (acyclic path decomposition number)* of and is denoted by
() (()). In this paper we initiate a study of the parameter and
determine the value of for some standard graphs. Further, we obtain
some bounds for and characterize graphs attaining the bounds. We
also prove that the difference between the parameters
and can be made arbitrarily large.