Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The $(\Delta,D)$ (degree/diameter) problem consists of finding the largest possible number of vertices $n$ among all the graphs with maximum degree $\Delta$ and diameter $D$.  consider the $(\Delta,D)$ problem for maximal planar bipartite graphs, that is, simple planar graphs in which every face is a quadrangle.  obtain that for the $(\Delta,2)$ problem, the number of vertices is $n=\Delta+2$; and for the $(\Delta,3)$ problem, $n= 3\Delta-1$ if $\Delta$ is odd and $n= 3\Delta-2$ if $\Delta$ is even. Then,  prove that, for the general case of the $(\Delta,D)$ problem, an upper bound on $n$ is approximately $3(2D+1)(\Delta-2)^{\lfloor D/2\rfloor}$, and another one is $C(\Delta-2)^{\lfloor D/2\rfloor}$ if $\Delta\geq D$ and $C$ is a sufficiently large constant. Our upper bounds improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs.  also give a lor bound on $n$ for maximal planar bipartite graphs, which is approximately $(\Delta-2)^{k}$ if $D=2k$, and $3(\Delta-3)^k$ if $D=2k+1$, for $\Delta$ and $D$ sufficiently large in both cases.
['Cristina Dalfó', 'Clemens Huemer', 'Julián Salas']
I am writing a research paper on a idea The degree/diameter problem in maximal planar bipartite graphs
[{"from": "human", "value": "I am writing a research paper on a idea The degree/diameter problem in maximal planar bipartite graphs"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The $(\\Delta,D)$ (degree/diameter) problem consists of finding the largest possible number of vertices $n$ among all the graphs with maximum degree $\\Delta$ and diameter $D$.\u00a0 consider the $(\\Delta,D)$ problem for maximal planar bipartite graphs, that is, simple planar graphs in which every face is a quadrangle.\u00a0 obtain that for the $(\\Delta,2)$ problem, the number of vertices is $n=\\Delta+2$; and for the $(\\Delta,3)$ problem, $n= 3\\Delta-1$ if $\\Delta$ is odd and $n= 3\\Delta-2$ if $\\Delta$ is even. Then,  prove that, for the general case of the $(\\Delta,D)$ problem, an upper bound on $n$ is approximately $3(2D+1)(\\Delta-2)^{\\lfloor D/2\\rfloor}$, and another one is $C(\\Delta-2)^{\\lfloor D/2\\rfloor}$ if $\\Delta\\geq D$ and $C$ is a sufficiently large constant. Our upper bounds improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs.  also give a lor bound on $n$ for maximal planar bipartite graphs, which is approximately $(\\Delta-2)^{k}$ if $D=2k$, and $3(\\Delta-3)^k$ if $D=2k+1$, for $\\Delta$ and $D$ sufficiently large in both cases."}]