irredundance chromatic number and gamma chromatic number of trees

A vertex subset $s$ of a graph $g = (v, e)$ is irredundant if every vertex in $s$ has a private neighbor with respect to $s$. an irredundant set $s$ of $g$ is maximal if, for any $v in v - s$, the set $s cup {v}$ is no longer irredundant. the lower irredundance number of $g$ is the minimum cardinality of a maximal irredundant set of $g$ and is denoted by $ir(g)$. a coloring $mathcal{c}$ of $g$ is said to be the irredundance coloring if there exists a maximal irredundant set $r$ of $g$ such that all the vertices of $r$ receive different colors. the minimum number of colors required for an irredundance coloring of $g$ is called the irredundance chromatic number of $g$, and is denoted by $chi_{i}(g)$. a coloring $mathcal{c}$ of $g$ is said to be the gamma coloring if there exists a dominating set $d$ of $g$ such that all the vertices of $d$ receive different colors. the minimum number of colors required for a gamma coloring of $g$ is called the gamma chromatic number of $g$, and is denoted by $chi_{gamma}(g)$. in this paper, we prove that every tree $t$ satisfies $chi_{i}(t) = ir(t)$ unless $t$ is a star. further, we prove that $gamma(t) leq chi_{gamma}(t) leq gamma(t) + 1$. we characterize all trees satisfying the upper bound.