A Note on ID-Colorings and Symmetric Colorings of Paths

A red–white coloring of a nontrivial connected graph [Formula: see text] is an assignment of red and white colors to the vertices of [Formula: see text]. Associated with each vertex [Formula: see text] of [Formula: see text] of diameter [Formula: see text] is a [Formula: see text]-vector, called the code of [Formula: see text], whose [Formula: see text]th coordinate is the number of red vertices at distance [Formula: see text] from [Formula: see text]. A red–white coloring of [Formula: see text] for which distinct vertices have distinct codes is called an ID-coloring of [Formula: see text]. A pair of distinct vertices on a path of order at least 2 are called partners if they have the same eccentricity. A red–white coloring of a path is called a symmetric coloring if partner vertices have the same color. In 2024, it was shown by Marcelo et al. that any red–white coloring of a path whose two leaves are red is an ID-coloring if it is not a symmetric coloring. In this paper, we prove the converse of this fact, and we use the result to establish a criterion to determine whether a red–white coloring of a path is an ID-coloring or not.