gives a better bound than Proposition \[prop2\](a) when $r \ge \sqrt{n/\log n}$ and a better bound than Proposition \[prop2\](b) when $\chi(G) \ge n/\log n$.

Trees
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In this section, we focus our attention on trees. We shall need the following notation. A vertex of degree one is called a *leaf* and its neighbor is called a *support vertex*. The set of support vertices of a tree $T$ is denoted by $S_T$, while the set of leaves by $L_T$. A neighbor of a vertex $v$ that is a leaf we call a *leaf-neighbor* of $v$. A *strong support vertex* is a vertex adjacent to at least two leaves. The *corona* of a graph $H$, denoted by ${{\rm cor}}(H)$, is the graph of order $2|V(H)|$ obtained from $H$ by attaching a leaf to each vertex of $H$. We note that every vertex of ${{\rm cor}}(H)$ is a leaf or is a support vertex with exactly one leaf-neighbor.

If we restrict our attention to trees, then the bound in Theorem \[thm\_n2\] can be improved significantly. For this purpose, we recall that a classical result of Ore [@o9] established that if $G$ is a graph of order $n$ with no isolated vertex, then $\gamma(G) 