LATIN 2018: LATIN 2018: Theoretical Informatics pp 399-412

# Shifting the Phase Transition Threshold for Random Graphs Using Degree Set Constraints

• Sergey Dovgal
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)

## Abstract

We show that by restricting the degrees of the vertices of a graph to an arbitrary set $$\varDelta$$, the threshold point $$\alpha (\varDelta )$$ of the phase transition for a random graph with $$n$$ vertices and $$m = \alpha (\varDelta ) n$$ edges can be either accelerated (e.g., $$\alpha (\varDelta ) \approx 0.381$$ for $$\varDelta = \{0,1,4,5\}$$) or postponed (e.g., $$\alpha (\{ 2^0, 2^1, \cdots , 2^k, \cdots \}) \approx 0.795$$) compared to a classical Erdős–Rényi random graph with $$\alpha (\mathbb Z_{\ge 0}) = \tfrac{1}{2}$$. 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 $$\alpha (\varDelta ) n$$. We investigate these probabilities and also different graph statistics inside the critical window of transition (diameter, longest path and circumference of a complex component).

## Notes

### Acknowledgements

We would like to thank Fedor Petrov for his help with a proof of technical condition for saddle-point analysis, Élie de Panafieu, Lutz Warnke, and several anonymous referees for their valuable remarks.

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