On bounds for some graph invariants

Abstract

Let $G$ be a graph without isolated vertices and let $α(G)$ be its stability number and $τ(G)$ its covering number. The {\it $α_{v}$-cover} number of a graph, denoted by $α_{v}(G)$, is the maximum natural number $m$ such that every vertex of $G$ belongs to a maximal independent set with at least $m$ vertices. In the first part of this paper we prove that $α(G)\leq τ(G)[1+α(G)-α_{v}(G)]$. We also discuss some conjectures analogous to this theorem. In the second part we give a lower bound for the number of edges of a graph $G$ as a function of the stability number $α(G)$, the covering number $τ(G)$ and the number of connected components $c(G)$ of $G$. Namely, let $α$ and $τ$ be two natural numbers and let $$ Γ(α,τ)= \min{\sum_{i=1}^α\bin{z_i}{2} | z_1+...+z_α= α+τ{and} z_i \geq 0 \forall i=1,..., α}. $$ Then if $G$ is any graph, we have: $$ |E(G)| \geq α(G)-c(G)+ Γ(α(G), τ(G)). $$