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What is the optimal upper bound of $|T_1|+|T_2|+|T_3|$ if $T_1, T_2, T_3$ are trees covering a planar graph

By a classical theorem of Nash-Williams, the edges of every connected $n$-vertex planar graph can be covered by three trees $T_1,T_2$ and $T_3$. Does anyone know of any results from an article or a ...
Xin Zhang's user avatar
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Estimates for the drop in the sum of all pairwise weighted distances effected by a decrease of the weight of an edge

Consider simple connected undirected graphs $G = (V, E)$ equipped with a function $w\colon E\rightarrow \{x\in \mathbb{R}\colon x\geq 0\}$. Define a function $d_{G,w}\colon V\times V\rightarrow\...
Lwins's user avatar
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