Characterization of \(P_3\cup P_2\)-Equipackable Graphs with \(3m (m\ge 1)\) Edges

Let H be a subgraph of graph G. If the edge set of G can be partitioned into edge-disjoint copies of H, possibly with some remainder edges, then the partition is called an H-packing in G. A maximal H-packing in G is defined as one where the remainder edges do not contain any copies of H. A maximal H-packing of G is considered maximum if the edge set E(G) cannot be partitioned into an H-packing containing more copies of H. A graph G is referred to as being H-equipackable if every maximal H-packing in G is also maximum H-packing. In this paper, we provide a characterization for all \(P_3\cup P_2\) $P_3\cup P_2$ -equipackable graphs with \(3\,m(m\ge 1)\) $3m(m\ge 1)$ edges, where \(P_3\cup P_2\) $P_3\cup P_2$ represents the union of two disjoint paths with 3 vertices and 2 vertices.