We introduce an algorithm that constructs a random graph with prescribed degree sequence together with a depth first exploration of it. In the so-called supercritical regime where the graph contains a giant component, we prove that the renormalized contour process of the Depth First Search Tree has a deterministic limiting profile that we identify. The proof goes through a detailed analysis of the evolution of the empirical degree distribution of unexplored vertices. This evolution is driven by an infinite system of differential equations which has a unique and explicit solution. As a byproduct, we deduce the existence of a macroscopic simple path and get a lower bound on its length.
The first author would like to thank the ANR grants MALIN (Projet- ANR-16-CE93-0003) and PPPP (Projet-ANR-16-CE40-0016) for their financial support. The other three authors would like to thank the ANR grant ProGraM (Projet-ANR-19-CE40-0025) for its financial support. G.F. and L.M. also acknowledge the support of the Labex MME-DII (ANR11-LBX-0023-01).
The authors are pleased to thank warmly an anonymous referee for its careful reading, suggestions, and pointing out a mistake in the original manuscript.
"Depth first exploration of a configuration model." Electron. J. Probab. 27 1 - 27, 2022. https://doi.org/10.1214/22-EJP762