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.
Highlights
Graph decomposition problems rank among the most prominent areas of research in graph theory and combinatorics and further it has numerous applications in various fields such as networking, block designs, and bioinformatics.A decomposition of a graph G is a collection of edgedisjoint subgraphs H1, H2, . . . , Hr of G such that every edge of G belongs to exactly one Hi
It is obvious that every graph admits a decomposition in which each subgraph Hi is either a path or a cycle
This paper initiates a study of the parameter path decomposition number
Summary
Graph decomposition problems rank among the most prominent areas of research in graph theory and combinatorics and further it has numerous applications in various fields such as networking, block designs, and bioinformatics. It is obvious that every graph admits a decomposition in which each subgraph Hi is either a path or a cycle. In this connection, Erdös asked what is the minimum number of paths into which every connected graph on n vertices can be decomposed and Gallai conjectured that this number is at most ⌈n/2⌉ as stated below. A good number of research articles have been published in which Gallai’s is the focus of study and still this conjecture remains unsettled for more than 30 years. A graph G on n vertices (not necessarily connected) can be decomposed into ⌊n/2⌋ paths and cycles. Gallai’s conjecture and Theorem 1 motivate the following definition. This paper initiates a study of the parameter path decomposition number
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