Cycle saturation in random graphs

TL;DR

This paper investigates the saturation number of cycles in random graphs, establishing asymptotic bounds for cycles of length 4 and 5 or more, revealing new behaviors in probabilistic graph saturation.

Contribution

It provides the first asymptotic results for the cycle saturation number in Erdős–Rényi random graphs for cycles of length four and greater.

Findings

For cycles of length m ≥ 5, the saturation number is whp n + Θ(n/ln n).

For C4, the saturation number is between 1.5n and cn, with c depending on p.

Specifically, for p=1/2, the saturation number for C4 is at most (27/14)n + o(n).

Abstract

For a fixed graph $F,$ the minimum number of edges in an edge-maximal $F$-free subgraph of $G$ is called the $F$-saturation number. The asymptotics of the $F$-saturation number of the binomial random graph $G(n,p)$ for constant $p\in(0,1)$ is known for complete graphs $F=K_m$ and stars $F=K_{1,m}.$ This paper is devoted to the case when the pattern graph $F$ is a simple cycle $C_m.$ We prove that, for $m\geqslant 5,$ whp $\mathrm{sat}\left(G\left(n,p\right),C_m\right) = n+\Theta\left(\frac{n}{\ln n}\right).$ Also we find $c=c(p)$ such that whp $\frac{3}{2}n(1+o(1))\leqslant\mathrm{sat}\left(G\left(n,p\right),C_4\right)\leqslant cn(1+o(1)).$ In particular, whp $\mathrm{sat}\left(G\left(n,\frac{1}{2}\right),C_4\right)\leqslant\frac{27}{14}n(1+o(1)).$