Path Decomposition Number of Certain Graphs

Abstract

Let G be a simple, finite and connected graph. A graph is said to be decomposed into subgraphs H1 and H2 which is denoted by G = H1 ⊕H2, if G is the edge disjoint union of H1 and H2. Assume that G = H1 ⊕ H2 ⊕ · · · ⊕ Hk and if each Hi, 1 ≤ i ≤ k, is a path or cycle in G, then the collection of edge-disjoint subgraphs of G denoted by ψ is called a path decomposition of G. If each Hi is a path in G then ψ is called an acyclic path decomposition of G. The minimum cardinality of a path decomposition of G, denoted by π(G), is called the path decomposition number and the minimum cardinality of an acyclic path decomposition of G, denoted by πa(G), is called the acyclic path decomposition number of G. In this paper, we determine path decomposition number for a number of graphs in particular, the Cartesian product of graphs. We also provided bounds for π(G) and πa(G) for these graphs. Mathematics Subject Classification: 05C70.