Shifting the Phase Transition Threshold for Random Graphs and 2-SAT using Degree Constraints

We show that by restricting the degrees of the vertices of a graph to an arbitrary set $\Delta$, the threshold point $\alpha(\Delta)$ of the phase transition for a random graph with $n$ vertices and $m = \alpha(\Delta) n$ edges can be either accelerated (e.g., $\alpha(\Delta) \approx 0.381$ for $\Delta = \{0,1,4,5\}$) or postponed (e.g., $\alpha(\{ 2^0, 2^1, \cdots, 2^k, \cdots \}) \approx 0.795$) compared to a classical Erd\H{o}s--R\'{e}nyi random graph with $\alpha(\mathbb Z_{\geq 0}) = \tfrac12$. In particular, we prove that the probability of graph being nonplanar and the probability of having a complex component, goes from $0$ to $1$ as $m$ passes $\alpha(\Delta) n$. We investigate these probabilities and also different graph statistics inside the critical window of transition (diameter, longest path and circumference of a complex component).