Trees with small b-chromatic index

In a recent article [5], the authors claim that the distance between the b-chromatic index of a tree and a known upper bound is at most 1. At the same time, in [7] the authors claim to be able to construct a tree where this difference is bigger than 1. However, the given example was disconnected, i.e., actually consisted of a forest. Here, we slightly modify their construction in order to produce trees, thus getting that indeed the difference between the b-chromatic index of trees and the known upper bound can be arbitrarily large. We also point out the mistake made in [5]. Given a proper coloring ψ : V (G) → {1, · · · , k} of G, we say that color i ∈ {1, · · · , k} is realized in ψ if there exists u ∈ V (G) such that ψ(u) = i and ψ(N [u]) = {1, · · · , k}; we also say that u realizes color i in ψ, and that u is a b-vertex of ψ. A b-coloring of the vertices of a graph is a proper coloring that realizes every color. Observe that if ψ does not realize color i, then each vertex u in color class i can have its color modified to some j ∈ {1, · · · , k}\ψ(N [u]). This produces a proper color that uses fewer colors, which implies that any coloring with χ(G) colors must be a b-coloring. Therefore, we are actually interested in the worst case scenario of such a coloring, and define the b-chromatic number of G as the maximum integer b(G) for which G has a b-coloring with b(G) colors. This problem was introduced by Irving and Manlove in [4], where they also showed that computing b(G) is NP-hard in general and polynomial-time solvable for trees. The problem is still NP-hard when restricted to bipartite graphs [6], chordal distance-hereditary graphs [3], and line graphs [1]. In their seminal paper, Irving and Manlove also introduced an important upper bound for this metric. Given a b-coloring with k colors, observe that there must exist at least k vertices in G with degree at least k − 1 (namely, the b-vertices). Therefore, if m(G) is defined as being the largest integer k such that G has at least k vertices with degree at least k − 1, then if follows that: