Multi-Decomposition of Graphs into Paths and \(Y\)-Trees of Order Five

Let \(K_n\), \(P_n\), and \(Y_n\) respectively denote a complete graph, a path, and a \(Y\)-tree on \(n\) vertices, and let \(K_{m,n}\) denote a complete bipartite graph with \(m\) and \(n\) vertices in its parts. Graph decomposition is the process of breaking down a graph into a collection of edge-disjoint subgraphs. A graph \(G\) has a \((H_1, H_2)\)-multi-decomposition if it can be decomposed into \(\alpha \geq 0\) copies of \(H_1\) and \(\beta \geq 0\) copies of \(H_2\), where \(H_1\) and \(H_2\) are subgraphs of \(G\). In this paper, we derive the necessary and sufficient conditions for the \((P_5, Y_5)\)-multi-decomposition of \(K_n\) and \(K_{m,n}\).