Multiple whole-genome alignment is a challenging problem in bioinformatics. Despite many successes, current methods are not able to keep up with the growing number, length, and complexity of assembled genomes, especially when computational resources are limited. Approaches based on compacted de Bruijn graphs to identify and extend anchors into locally collinear blocks have potential for scalability, but current methods do not scale to mammalian genomes. We present an algorithm, SibeliaZ-LCB, for identifying collinear blocks in closely related genomes based on analysis of the de Bruijn graph. We further incorporate this into a multiple whole-genome alignment pipeline called SibeliaZ. SibeliaZ shows run-time improvements over other methods while maintaining accuracy. On sixteen recently-assembled strains of mice, SibeliaZ runs in under 16 hours on a single machine, while other tools did not run to completion for eight mice within a week. SibeliaZ makes a significant step towards improving scalability of multiple whole-genome alignment and collinear block reconstruction algorithms on a single machine.