Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let G denote a multigraph with edge set E(G), let µ(G) denote the maximum edge multiplicity in G, and let Pk denote the path on k vertices. Heinrich et al.(1999) shod that P4 decomposes a connected 4-regular graph G if and only if |E(G)| is divisible by 3.  show that P4 decomposes a connected 4-regular multigraph G with µ(G) ≤2 if and only if no 3 vertices of G induce more than 4 edges and |E(G)| is divisible by 3. Oksimets (2003) proved that for all integers k ≥3, P4 decomposes a connected 2k-regular graph G if and only if |E(G)| is divisible by 3.  prove that for all integers k ≥2, the problem of determining if P4 decomposes a (2k + 1)-regular graph is NP-Complete. El-Zanati et al.(2014) shod that for all integers k ≥1, every 6k-regular multigraph with µ(G) ≤2k has a P4-decomposition.  show that unless P = NP, this result is best possible with respect to µ(G) by proving that for all integers k ≥3 the problem of determining if P4 decomposes a 2k-regular multigraph with µ(G) ≤⌊2k / 3 ⌋+ 1 is NP-Complete.
['Ajit A. Diwan', 'Justine E. Dion', 'David Joshua Mendell', 'Michael J. Plantholt', 'Shailesh K. Tipnis']
I am writing a research paper on a idea The complexity of $P$ 4 -decomposition of regular graphs and multigraphs
[{"from": "human", "value": "I am writing a research paper on a idea The complexity of $P$ 4 -decomposition of regular graphs and multigraphs"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let G denote a multigraph with edge set E(G), let \u00b5(G) denote the maximum edge multiplicity in G, and let Pk denote the path on k vertices. Heinrich et al.(1999) shod that P4 decomposes a connected 4-regular graph G if and only if |E(G)| is divisible by 3.  show that P4 decomposes a connected 4-regular multigraph G with \u00b5(G) \u22642 if and only if no 3 vertices of G induce more than 4 edges and |E(G)| is divisible by 3. Oksimets (2003) proved that for all integers k \u22653, P4 decomposes a connected 2k-regular graph G if and only if |E(G)| is divisible by 3.  prove that for all integers k \u22652, the problem of determining if P4 decomposes a (2k + 1)-regular graph is NP-Complete. El-Zanati et al.(2014) shod that for all integers k \u22651, every 6k-regular multigraph with \u00b5(G) \u22642k has a P4-decomposition.  show that unless P = NP, this result is best possible with respect to \u00b5(G) by proving that for all integers k \u22653 the problem of determining if P4 decomposes a 2k-regular multigraph with \u00b5(G) \u2264\u230a2k\u200a/\u200a3 \u230b+ 1 is NP-Complete."}]