The 3-rainbow index of graph operations

TL;DR

This paper investigates the 3-rainbow index of graphs under various operations like Cartesian, strong, and lexicographic products, providing bounds and conditions for equality, with constructive proofs and sharp bounds.

Contribution

It establishes bounds for the 3-rainbow index of graph products and other operations, including conditions for equality and constructive proofs for sharp bounds.

Findings

Upper bound for rx_3 of Cartesian product graphs

Sharp bounds for rx_3 of lexicographic product graphs

Relationships between rx_3 and basic graph operations

Abstract

A tree $T$, in an edge-colored graph $G$, is called {\em a rainbow tree} if no two edges of $T$ are assigned the same color. A {\em $k$-rainbow coloring}of $G$ is an edge coloring of $G$ having the property that for every set $S$ of $k$ vertices of $G$, there exists a rainbow tree $T$ in $G$ such that $S\subseteq V(T)$. The minimum number of colors needed in a $k$-rainbow coloring of $G$ is the {\em $k$-rainbow index of $G$}, denoted by $rx_k(G)$. Graph operations, both binary and unary, are an interesting subject, which can be used to understand structures of graphs. In this paper, we will study the $3$-rainbow index with respect to three important graph product operations (namely cartesian product, strong product, lexicographic product) and other graph operations. In this direction, we firstly show if $G^*=G_1\Box G_2\cdots\Box G_k$ ($k\geq 2$), where each $G_i$ is connected, then $rx_3(G^*)\leq \sum_{i=1}^{k} rx_3(G_i)$. Moreover, we also present a condition and show the above equality holds if every graph $G_i (1\leq i\leq k)$ meets the condition. As a corollary, we obtain an upper bound for the 3-rainbow index of strong product. Secondly, we discuss the 3-rainbow index of the lexicographic graph $G[H]$ for connected graphs $G$ and $H$. The proofs are constructive and hence yield the sharp bound. Finally, we consider the relationship between the 3-rainbow index of original graphs and other simple graph operations : the join of $G$ and $H$, split a vertex of a graph and subdivide an edge.