Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: By $\delta$ and $w_k$ denote the minimum degree and minimum degree-sum (ight) of a $k$-vertex path in a given graph, respectively. For every 3-polytope, $w_2\le13$ (Kotzig, 1955) and $w_3\le21$ (Ando, Iwasaki, Kaneko, 1993), where both bounds are sharp. For every 3-polytope with $\delta\ge4$,  have sharp bounds $w_2\le11$ (Lebesgue, 1940) and $w_3\le17$ (Borodin, 1997).  Madaras (2000) proved that every triangulated 3-polytope with $\delta\ge4$ satisfies $w_4\le31$ and constructed such a 3-polytope with $w_4=27$.   improve the Madaras bound $w_4\le31$ to the sharp bound $w_4\le27$.
['Oleg V. Borodin', 'Anna O. Ivanova']
I am writing a research paper on a idea Every Triangulated 3-Polytope of Minimum Degree 4 has a 4-Path of Weight at Most 27
[{"from": "human", "value": "I am writing a research paper on a idea Every Triangulated 3-Polytope of Minimum Degree 4 has a 4-Path of Weight at Most 27"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: By $\\delta$ and $w_k$ denote the minimum degree and minimum degree-sum (ight) of a $k$-vertex path in a given graph, respectively.\u00a0For every 3-polytope, $w_2\\le13$ (Kotzig, 1955) and $w_3\\le21$ (Ando, Iwasaki, Kaneko, 1993), where both bounds are sharp. For every 3-polytope with $\\delta\\ge4$,  have sharp bounds $w_2\\le11$ (Lebesgue, 1940) and $w_3\\le17$ (Borodin, 1997).  Madaras (2000) proved that every triangulated 3-polytope with $\\delta\\ge4$ satisfies $w_4\\le31$ and constructed such a 3-polytope with $w_4=27$.   improve the Madaras bound $w_4\\le31$ to the sharp bound $w_4\\le27$."}]