Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let G be a simple graph of order n. The path cover number μ(G )i s defined to be the minimum number of vertex disjoint paths required to cover the vertices of G. Ore proved that in general μ(G) ≤ max{1 ,n − σ2(G)}.  conjecture that if G is k-regular, then μ(G) ≤ n k+1 and  prove this for k ≤ 5.
['Colton Magnant', 'Daniel M. Martin']
I am writing a research paper on a idea A note on the path cover number of regular graphs
[{"from": "human", "value": "I am writing a research paper on a idea A note on the path cover number of regular graphs"}, {"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 be a simple graph of order n. The path cover number \u03bc(G )i s defined to be the minimum number of vertex disjoint paths required to cover the vertices of G. Ore proved that in general \u03bc(G) \u2264 max{1 ,n \u2212 \u03c32(G)}.  conjecture that if G is k-regular, then \u03bc(G) \u2264 n k+1 and  prove this for k \u2264 5."}]