Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Given graphs $G$ and $H$ and a positive number $b$, a  ighted $(H,b)$-decomposition  of $G$ is a partition of the edge set of $G$ such that each part is either a single edge or forms an $H$-subgraph.  assign a ight of $b$ to each $H$-subgraph in the decomposition and a ight of 1 to single edges. The total ight of the decomposition is the sum of the ights of all elements in the decomposition. Let $\phi(n,H,b)$ be the the smallest number such that any graph $G$ of order $n$ admits an $(H,b)$-decomposition with ight at most $\phi(n,H,b)$. The value of the function $\phi(n,H,b)$ when $b=1$ was determined, for large $n$, by Pikhurko and Sousa [ Minimum $H$-Decompositions of Graphs , Journal of Combinatorial Theory, B,  97  (2007), 1041–1055.]  Here  determine the asymptotic value of $\phi(n,H,b)$ for any fixed bipartite graph $H$ and any value of $b$ as $n$ tends to infinity.
['Teresa Sousa']
I am writing a research paper on a idea Minimum Weight $H$-Decompositions of Graphs: The Bipartite Case
[{"from": "human", "value": "I am writing a research paper on a idea Minimum Weight $H$-Decompositions of Graphs: The Bipartite Case"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Given graphs $G$ and $H$ and a positive number $b$, a  ighted $(H,b)$-decomposition  of $G$ is a partition of the edge set of $G$ such that each part is either a single edge or forms an $H$-subgraph.  assign a ight of $b$ to each $H$-subgraph in the decomposition and a ight of 1 to single edges. The total ight of the decomposition is the sum of the ights of all elements in the decomposition. Let $\\phi(n,H,b)$ be the the smallest number such that any graph $G$ of order $n$ admits an $(H,b)$-decomposition with ight at most $\\phi(n,H,b)$. The value of the function $\\phi(n,H,b)$ when $b=1$ was determined, for large $n$, by Pikhurko and Sousa [ Minimum $H$-Decompositions of Graphs , Journal of Combinatorial Theory, B,  97  (2007), 1041\u20131055.]  Here  determine the asymptotic value of $\\phi(n,H,b)$ for any fixed bipartite graph $H$ and any value of $b$ as $n$ tends to infinity."}]