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The global structure of the minimal spanning tree (MST) is expected to beuniversal for a large class of underlying random discrete structures. However,very little is known about the intrinsic geometry of MSTs of most standardmodels, and so far the scaling limit of the MST viewed as a metric measurespace has only been identified in the case of the complete graph [5]. In thiswork, we show that the MST constructed by assigning i.i.d. continuousedge-weights to either the random (simple) $3$-regular graph or the $3$-regularconfiguration model on $n$ vertices, endowed with the tree distance scaled by$n^{-1/3}$ and the uniform probability measure on the vertices, converges indistribution with respect to Gromov-Hausdorff-Prokhorov topology to a randomcompact metric measure space. Further, this limiting space has the same law asthe scaling limit of the MST of the complete graph identified in [5] up to ascaling factor of $6^{1/3}$. Our proof relies on a novel argument that proceedsvia a comparison between a $3$-regular configuration model and the largestcomponent in the critical Erd\H{o}s-R\'enyi random graph. The techniques ofthis paper can be used to establish the scaling limit of the MST in the settingof general random graphs with given degree sequences provided two additionaltechnical conditions are verified.

Geometry of the minimal spanning tree of a random $3$-regular graph