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

TL;DR

This paper proves new inequalities relating core, corona, and critical sets in graphs, confirming recent conjectures and establishing bounds involving odd cycles and maximum independent sets.

Contribution

It establishes bounds on the sizes of core, corona, nucleus, and diadem sets in graphs, confirming two recent conjectures and linking these to the number of odd cycles.

Findings

Proved that |corona(G)| + |core(G)| ≤ 2α(G) + k.

Confirmed that |nucleus(G)| + |diadem(G)| ≤ 2α(G).

Established a chain of inequalities involving these sets and α(G).

Abstract

Let $\alpha(G)$ 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 \[ \left|\text{corona}(G)\right|+\left|\text{core}(G)\right|\le2\alpha(G)+k, \] \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 \[ \left|\text{nucleus}(G)\right|+\left|\text{diadem}(G)\right|\le2\alpha(G), \] \noindent thereby confirming another conjecture (Levit--Mandrescu 2014). As an application of these facts, we obtain a chain of inequalities \[ \left|\text{nucleus}(G)\right|+\left|\text{diadem}(G)\right|\le2\alpha(G)\le\left|\text{corona}(G)\right|+\left|\text{core}(G)\right|\le2\alpha(G)+k. \] \noindent The paper concludes with a collection of related open problems.