The (vertex-)monochromatic index of a graph

TL;DR

This paper introduces and analyzes the concepts of the $k$-monochromatic index and vertex-monochromatic index of graphs, providing exact formulas, complexity results, and Nordhaus-Gaddum-type bounds for these new graph invariants.

Contribution

The paper defines the $k$-monochromatic index and vertex-monochromatic index, proves exact values for the former, establishes NP-completeness for the decision problem of the latter, and derives Nordhaus-Gaddum bounds.

Findings

$mx_k(G)=|E(G)|-|V(G)|+2$ for all connected graphs and $3\leq k o |V(G)|$

Deciding whether $mvx_k(G)\geq L$ is NP-complete for all $2 ext o |V(G)|$

Derived Nordhaus-Gaddum-type inequalities for $mvx_k(G)$

Abstract

A tree $T$ in an edge-colored graph $H$ is called a \emph{monochromatic tree} if all the edges of $T$ have the same color. For $S\subseteq V(H)$, a \emph{monochromatic $S$-tree} in $H$ is a monochromatic tree of $H$ containing the vertices of $S$. For a connected graph $G$ and a given integer $k$ with $2\leq k\leq |V(G)|$, the \emph{$k$-monochromatic index $mx_k(G)$} of $G$ is the maximum number of colors needed such that for each subset $S\subseteq V(G)$ of $k$ vertices, there exists a monochromatic $S$-tree. In this paper, we prove that for any connected graph $G$, $mx_k(G)=|E(G)|-|V(G)|+2$ for each $k$ such that $3\leq k\leq |V(G)|$. A tree $T$ in a vertex-colored graph $H$ is called a \emph{vertex-monochromatic tree} if all the internal vertices of $T$ have the same color. For $S\subseteq V(H)$, a \emph{vertex-monochromatic $S$-tree} in $H$ is a vertex-monochromatic tree of $H$ containing the vertices of $S$. For a connected graph $G$ and a given integer $k$ with $2\leq k\leq |V(G)|$, the \emph{$k$-monochromatic vertex-index $mvx_k(G)$} of $G$ is the maximum number of colors needed such that for each subset $S\subseteq V(G)$ of $k$ vertices, there exists a vertex-monochromatic $S$-tree. We show that for a given a connected graph $G$, and a positive integer $L$ with $L\leq |V(G)|$, to decide whether $mvx_k(G)\geq L$ is NP-complete for each integer $k$ such that $2\leq k\leq |V(G)|$. We also obtain some Nordhaus-Gaddum-type results for the $k$-monochromatic vertex-index.