On Characterizations of Trees Having Large \((2,0)\)-Chromatic Numbers

For a nontrivial connected graph \( G \), let \( c: V(G) \to \mathbb{Z}_2 \) be a vertex coloring of \( G \) where \( c(v) \neq 0 \) for at least one vertex \( v \) of \( G \). Then the coloring \( c \) induces a new coloring \( \sigma: V(G) \to \mathbb{Z}_2 \) defined by \( \sigma(v) = \sum_{u \in N[v]} c(u) \), where \( N[v] \) is the closed neighborhood of \( v \) and addition is performed in \( \mathbb{Z}_2 \). If \( \sigma(v) = 0 \in \mathbb{Z}_2 \) for every vertex \( v \) in \( G \), then the coloring \( c \) is called a modular monochromatic \( (2, 0) \)-coloring of \( G \). A graph \( G \) having a modular monochromatic \( (2, 0) \)-coloring is a monochromatic \( (2, 0) \)-colorable graph. The minimum number of vertices colored 1 in a modular monochromatic \( (2, 0) \)-coloring of \( G \) is the \( (2, 0) \)-chromatic number \( \chi_{(2,0)}(G) \) of \( G \). A monochromatic \( (2, 0) \)-colorable graph \( G \) of order \( n \) is \( (2, 0) \)-extremal if \( \chi_{(2,0)}(G) = n \). It is known that a tree \( T \) is \( (2,0) \)-extremal if and only if every vertex of \( T \) has odd degree. In this work, we characterize all trees of order \( n \) having \( (2,0) \)-chromatic number \( n-1 \), \( n-2 \), or \( n-3 \), and investigate the structures of connected graphs having large \( (2, 0) \)-chromatic numbers.