Inequalities Involving Core, Corona, and Critical Sets in General Graphs

Abstract

Let $α(G)$ denote the cardinality of a maximum independent set. An independent set $I$ of $G$ is critical if $\left|I\right|-\left|N(I)\right|\ge\left|J\right|-\left|N(J)\right|$ for every independent set $J$ of $G$. Let $\text{core}(G)$ and $\text{corona}(G)$ be the intersection/union of all maximum independent sets of $G$. Let $\text{ker}(G)$ and $\text{diadem}(G)$ be the intersection/union of all critical independent sets of $G$. In this paper we prove that \[ \left|\text{corona}(G)\right|+\left|\text{core}(G)\right|\le2α(G)+k, \] \noindent where $k$ is the number of vertex-distinct odd cycles in $G$, thus confirming a recent conjecture in the area. Moreover, we prove that \[ \left|\text{nucleus}(G)\right|+\left|\text{diadem}(G)\right|\le2α(G), \] \noindent thereby confirming another conjecture (Levit--Mandrescu 2014). As an application of these facts, we obtain a chain of inequalities \[ \left|\text{nucleus}(G)\right|+\left|\text{diadem}(G)\right|\le2α(G)\le\left|\text{corona}(G)\right|+\left|\text{core}(G)\right|\le2α(G)+k. \] \noindent The paper concludes with a collection of related open problems.

Figures (2)

• Figure 1: In this example, the vertices in $\text{corona}(G)$ are shown in blue. Note that $\left|\text{corona}(G)\right|+\left|\text{ker}(G)\right|=8+2=10<2\cdot5+2=2\alpha(G)+2,$ moreover $2$ is the number of odd cycles in $G$.
• Figure 2: Illustration of the proof of \ref{['mainlema2']}.

Theorems & Definitions (22)

• Conjecture 1.1: kevinBAB
• Conjecture 1.2: levit2014critical
• Theorem 1.3
• Theorem 3.1: levit2012vertices
• Theorem 3.2: jarden2017two
• Theorem 3.3
• proof
• Remark 3.4
• Corollary 3.5
• Theorem 3.6: berge2005some