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Suppose for a graph $G=(V,E)$ and a spanning tree T of G, $\Delta(T)$ is the largest degree of a vertex in T, and let $\Delta^*$ be the smallest such quantity over all spanning trees of $G$.

We have the following local search procedure which can changes spanning tree $T$ into a different spanning tree $T'$: We find an edge $e$ not in $T$ and add it to $T$. This results in a cycle $C$ - call its vertices vertices $V(C)$. We then delete an edge in $C$ incident to a vertex in $V(C)$ with highest degree.

My question is this: if $\Delta(T) > \Delta^*$, can we always find an edge $e$ to add, such that the maximum degree of vertices in $V(C)$ is strictly less in $T'$ than it is in $T$?

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  • $\begingroup$ How about just using a greedy approach, modifying Prim's algorithm to add an edge according to minimal degree of the origin vertex? I suspect that that will give minimal maximal degree... $\endgroup$ – vonbrand Nov 22 '15 at 1:46
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    $\begingroup$ For editing and commenting on your own posts, you should register your account. When you do so, you can request SE staff to merge your registered account with this one. (cc @D.W.) $\endgroup$ – Raphael Nov 22 '15 at 8:48
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No, we cannot always find an edge $e$ to add when $\Delta(T) > \Delta^*$.

An counterexample to show local search to find minimum degree spanning tree might not https://graphonline.ru/en/?graph=qUQjSqORjZZwsRMY

The graph above was drawn at https://graphonline.ru/en

Here is an counterexample. Let $G$ be the 3-regular graph on the right side that has 14 vertices and 21 edges. Let $T$ be the spanning tree that consists of all the edges in bold. That is, $T$ has all the edges in $G$ except the innermost 7 edges and the edge at the top right. The graph on the left side shows a hamiltonian path of $G$. That is, $\Delta(T)=3$ and $\Delta^*=2$.

Let $e$ be an edge not in $T$. If we add $e$ to $T$, the resulting cycle $C$ contains at least two non-adjacent vertices of degree 3 in $T$. If we further delete any edge in $C$ incident to a vertex in $C$ with the highest degree 3, we obtain graph $T'$. There is at least one vertex of degree 3 in $T'$. That is, the maximum degree of vertices in $V(C)$ in $T'$ is 3.

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