On the number of maximal independent sets and maximal induced bipartite subgraphs in $K_4$-free graphs

TL;DR

This paper addresses counting maximal independent sets by size and maximal induced bipartite subgraphs in K4-free graphs. The authors improve prior bounds by proving mis_k(G) ≤ (4−η)^{(5−η)k−n}(5−η)^{n−(4−η)k} for small η and showing an upper bound of O((12−ν)^{n/4}) on the number of maximal induced bipartite subgraphs, with tight behavior linked to unions of K4. The approach is inductive on the number of vertices, complemented by a detailed structural decomposition around a fixed maximal independent set and a sequence of layers that bound the number of MIS; for the mibs bound they employ a reduction to counting through partitions and probabilistic transversal arguments and auxiliary graphs. The results extend Nielsen's and Eppstein's bounds to the K4-free setting and suggest analogous bounds for $K_s$-free graphs, offering new directions in the study of MIS in triangle-free graphs.

Abstract

Let $G$ be a $K_4$-free graph of order $n$ and let $k$ be an integer with $0\leq k\leq n$. We show the existence of positive constants $η$ and $ν$ such that $G$ has at most $(4-η)^{(5-η)k-n}(5-η)^{n-(4-η)k}$ maximal independent sets of order $k$ and at most $O\left((12-ν)^{\frac{n}{4}}\right)$ maximal induced bipartite subgraphs.

Figures (4)

• Figure 1: The plot shows upper bounds on $\frac{\ln({\rm mis}_k(G))}{n}$ as a function of $x=\frac{k}{n}$ for the range $\frac{k}{n}\in \left[\frac{1}{5},\frac{1}{3}\right]$, where $G$ is a graph of order $n$. The straight line segment for $\frac{k}{n}\in \left[\frac{1}{4},\frac{1}{3}\right]$ is $\left(4\frac{k}{n}-1\right)\ln(3)+\left(1-3\frac{k}{n}\right)\ln(4)$, which corresponds to Eppstein's bound \ref{['e2']}. The straight line segment for $\frac{k}{n}\in \left[\frac{1}{5},\frac{1}{4}\right]$ is $\left(5\frac{k}{n}-1\right)\ln(4)+\left(1-4\frac{k}{n}\right)\ln(5)$, which corresponds to Nielsen's bound mentioned above. The dotted curve is the smooth interpolation $x\mapsto x\ln\left(\frac{1}{x}\right)$, which coincides with Nielsen's bound from ni whenever $\frac{1}{x}$ is an integer. The bound \ref{['e5']} from Corollary \ref{['corollary1']} corresponds to a line through the two points on the dashed curve for $x=\frac{1}{4-\eta}\in \left(\frac{1}{4},\frac{1}{3}\right)$ and $x=\frac{1}{5-\eta}\in \left(\frac{1}{5},\frac{1}{4}\right)$, which slightly improves Eppstein's and Nielsen's bounds close to $x=\frac{1}{4}$. See also the proof of Corollary \ref{['corollary1']} and Figure \ref{['figcor1']}.
• Figure 2: The structure of $G$ induced by $I_0$.
• Figure 3: Let $\{ v,v'\}=\{ x_i,y_i\}$. The probability $q$ that $T$ contains $v$ but no neighbor of $v'$ depends on the number $d$ of neighbors of $v'$ outside of $V_i$ and their distribution. In the figure we illustrate --- from left to right ---- the four possible cases $d=0$, $d=1$, $d=2$ and both neighbors of $v'$ outside of $V_i$ are in the same $V_j$, and $d=2$ and the two neighbors of $v'$ outside of $V_i$ are in distinct $V_j$'s. In the first case, we have $d=0$, $v'$ has all its neighbors in $V_i$, $v$ lies in $T$ with probability $1/4$, and, hence, $q=\frac{1}{4}$. In the second case, we have $d=1$, $v$ lies in $T$ with probability $1/4$, the unique neighbor of $v'$ outside of $V_i$ lies outside of $T$ with probability $3/4$, and, by the independence of these events, $q=\frac{1}{4}\cdot \frac{1}{4}$. In the third case, $v$ lies in $T$ with probability $1/4$, $T$ contains none of the two neighbors of $v'$ in $V_j$ with probability $1/2$, and, by the independence of these events, $q=\frac{1}{4}\cdot \frac{1}{2}$. Finally, in the fourth case, $v$ lies in $T$ with probability $1/4$, each of the two neighbors of $v'$ outside of $V_i$ lie outside of $T$ with probability $3/4$, and, by the independence of these events, $q=\frac{1}{4}\cdot \frac{3}{4}\cdot \frac{3}{4}$. The initial factor $2$ in \ref{['eq']} reflects that $v$ may be $x_i$ or $y_i$.
• Figure 4: The plot shows upper bounds on $\frac{\ln({\rm mis}_k(G))}{n}$ as a function of $x=\frac{k}{n}$. As in Figure \ref{['fig0']}, two of the staight line segments correspond to Eppstein's bound \ref{['e2']} in $\left[\frac{1}{4},\frac{1}{3}\right]$ and to Nielsen's bound in $\left[\frac{1}{5},\frac{1}{4}\right]$. The horizontal line segments illustrates a bound as in \ref{['e4']} from Theorem \ref{['theorem1']} in $\left[\frac{1}{4},\frac{1+\varepsilon}{4}\right]$. Continuously increasing $\eta$ from $0$ (Nielsen's bound) towards $1$ (Eppstein's bound) yields the dashed line segment corresponding to \ref{['e5']} as in Corollary \ref{['corollary1']}. In fact, the plot illustrates $\eta=0.4$.

Theorems & Definitions (8)

• Theorem 1: Eppstein, refined
• Theorem 2
• Corollary 3
• Corollary 4
• proof : Proof of Theorem \ref{['theorem2']}
• proof : Proof of Theorem \ref{['theorem1']}
• proof : Proof of Corollary \ref{['corollary1']}
• proof : Proof of Corollary \ref{['corollary2']}