A Jump of the Saturation Number in Random Graphs?

Abstract

For graphs $G$ and $F$, the saturation number $\textit{sat}(G,F)$ is the minimum number of edges in an inclusion-maximal $F$-free subgraph of $G$. In 2017, Korándi and Sudakov initiated the study of saturation in random graphs. They showed that for constant $p\in (0,1)$, whp $\textit{sat}\left(G(n,p),K_s\right)=\left(1+o(1)\right)n\log_{\frac{1}{1-p}}n$. We show that for every graph $F$ and every constant $p\in (0,1)$, whp $\textit{sat}\left(G(n,p), F\right)=O(n\ln n)$. Furthermore, if every edge of $F$ belongs to a triangle, then the above is the right asymptotic order of magnitude, that is, whp $\textit{sat}\left(G(n,p),F\right)=Θ(n\ln n)$. We further show that for a large family of graphs $\mathcal{F}$ with an edge that does not belong to a triangle, which includes all the bipartite graphs, for every $F\in \mathcal{F}$ and constant $p\in(0,1)$, whp $\textit{sat}\left(G(n,p),F\right)=O(n)$. We conjecture that this sharp transition from $O(n)$ to $Θ(n\ln n)$ depends only on this property, that is, that for any graph $F$ with at least one edge that does not belong to a triangle, whp $\textit{sat}\left(G(n,p),F\right)=O(n)$. We further generalise the result of Korándi and Sudakov, and show that for a more general family of graphs $\mathcal{F}'$, including all complete graphs $K_s$ and all complete multipartite graphs of the form $K_{1,1,s_3,\ldots, s_{\ell}}$, for every $F\in \mathcal{F}'$ and every constant $p\in(0,1)$, whp $\textit{sat}\left(G(n,p),F\right)=\left(1+o(1)\right)n\log_{\frac{1}{1-p}}n$. Finally, we show that for every complete multipartite graph $K_{s_1, s_2, \ldots, s_{\ell}}$ and every $p\in \left[\frac{1}{2},1\right)$, $\textit{sat}\left(G(n,p),K_{s_1,s_2,\ldots,s_{\ell}}\right)=\left(1+o(1)\right)n\log_{\frac{1}{1-p}}n$.

Figures (4)

• Figure 1: An illustration of graphs satisfying property $(\ntriangleright)$.
• Figure 2: An illustration of the edge $\{u,v\}$ and the remaining graph $F\setminus\{u,v\}$.
• Figure 3: In dashed red line there is a missing edge $\{u,v\}$ which closes a copy of $F'\in \hat{\mathcal{F}}$. Together with its common neighbourhood in $A$ (coloured light green and blue), this closes a copy of $F\in\mathcal{F}$.
• Figure 4: An illustration of how the edge $\{x,v\}$, which is not induced by $B_2$, is completed by $H_{1}$. In this case, we consider $K_{2,3,3}$. In $H_{B_1}$ we have the vertices $v, v_1\in B_1$ and $v_2\in B_2$, with the edges $\{v,v_1\}$ and $\{v,v_2\}$. In the common neighbourhood in $A_1$ of $\{x,v,v_1,v_2\}$, we can find inside $H_{A_1}$ a copy of $K_{1,3}$. Note that with the edges of $H_1$ and the edge $\{x,v\}$, we now have a copy of $K_{2,3,3}$, with its parts being $\{v,u\}, \{v_1,v_2,x\}, \{u_1, u_2, u_3\}$.

Theorems & Definitions (46)

• Theorem 1.1: Theorem 1.1 in KS17
• Theorem 1.2: Theorem 2.2 in KS17
• Theorem 1
• Conjecture 1.3
• Theorem 2
• Theorem 3
• Remark 1.4
• Theorem 4
• Corollary 1.5
• Lemma 1.6