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∈V−S, the set S∪{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 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 χi(G). A coloring 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 χγ(G). In this paper, we prove that every tree T satisfies χi(T)=ir(T) unless T is a star. Further, we prove that γ(T)≤χγ(T)≤γ(T)+1. We characterize all trees satisfying the upper bound.