$\delta_k$-small sets in graphs

TL;DR

This paper introduces $ ext{delta}_k$-small sets in graphs, providing bounds for their size and decomposition, which in turn yield bounds for key graph invariants like clique, chromatic, and independence numbers.

Contribution

It defines $ ext{delta}_k$-small sets and derives bounds for their maximal size and minimal decomposition, linking these to classical graph parameters.

Findings

Bounds for $eta^{(k)}(G)$ and $ ext{phi}^{(k)}(G)$ are established.

Results relate $ ext{delta}_k$-small sets to clique, chromatic, and independence numbers.

The paper provides new inequalities connecting these parameters.

Abstract

Let $G$ be a simple $n$-vertex graph and $W\subseteq\V(G)$. We say that $W$ is a $\delta_k$-small set if $$ \sqrt[k]{\frac{\sum_{v\in W}d^k(v)}{\abs W}}\leq n-\abs W. $$ Let $\varphi^{(k)}(G)$ denote the smallest natural number $r$ such that $\V(G)$ decomposes into $r$ $\delta_k$-small sets, and let $\alpha^{(k)}(G)$ denote the maximal number of vertices in a $\delta_k$-small set of $G$. In this paper we obtain bounds for $\alpha^{(k)}(G)$ and $\varphi^{(k)}(G)$. Since $\varphi^{(k)}(G)\leq\omega(G)\leq\chi(G)$ and $\alpha(G)\leq\alpha^{(k)}(G)$, we obtain also bounds for the clique number $\omega(G)$, the chromatic number $\chi(G)$ and the independence number $\alpha(G)$.