[Paper Review] The minimum size of graphs with given rainbow index

Thomas Y. H. Liu|arXiv (Cornell University)|Sep 24, 2015

Graph theory and applications4 references3 citations

TL;DR

This paper investigates the minimum size of connected graphs with a given $k$-rainbow index, focusing on $t(n,3,\ell)$, the smallest number of edges in an $n$-vertex graph where every 3-vertex subset admits a rainbow tree under some edge coloring. It establishes exact values and tight upper bounds for $t(n,3,\ell)$ across various $\ell$, including $t(n,3,3) \leq \lfloor n^2/4 \rfloor$ for even $n$ and $t(n,3,\ell) \leq 2n - \ell - 1$ for $n/2 \leq \ell \leq n-3$, with constructions using rose graphs and complete bipartite graphs.

ABSTRACT

The concept of $k$-rainbow index $rx_k(G)$ of a connected graph $G$, introduced by Chartrand, Okamoto and Zhang, is a natural generalization of the rainbow connection number. Let $t(n,k,\ell)$ denote the minimum size of a connected graph $G$ of order $n$ with $rx_k(G)\leq \ell$, where $2\leq \ell\leq n-1$ and $2\leq k\leq n$. In this paper, we obtain some exact values and some upper bounds for $t(n,k,\ell)$.

Motivation & Objective

• To determine the minimum number of edges in a connected graph of order $n$ such that its $k$-rainbow index is at most $\ell$, denoted $t(n,k,\ell)$.
• To focus specifically on the case $k=3$ and derive exact values and tight upper bounds for $t(n,3,\ell)$ across different values of $\ell$.
• To construct explicit families of graphs—such as complete bipartite graphs and rose graphs—that achieve the bounds and verify their $3$-rainbow index properties.
• To generalize prior results on rainbow connection and $k$-rainbow index by establishing a systematic framework for edge-minimal graphs with bounded $k$-rainbow index.

Proposed method

• Define $t(n,k,\ell)$ as the minimum number of edges in a connected $n$-vertex graph with $rx_k(G) \leq \ell$, generalizing the rainbow connection number.
• Use known results on $k$-rainbow index for trees, cycles, unicyclic graphs, and complete bipartite graphs as foundational tools.
• Construct specific edge-colored graphs—such as $K_{n/2,n/2}$ for even $n$, and $(n-\ell,3)$-rose graphs joined to paths—for which the $3$-rainbow index is bounded by $\ell$, proving upper bounds.
• Provide explicit edge-colorings using $\ell$ colors and verify that every 3-vertex subset admits a rainbow $S$-tree via case analysis on vertex distribution.
• Leverage structural properties: for $\ell \geq n/2$, use a path of length $2\ell - n$ attached to a rose graph with $n-\ell$ petals to achieve $t(n,3,\ell) \leq 2n - \ell - 1$.
• For $k=n-1$, use $K_{2,n-2}$ with a carefully assigned coloring to show $t(n,n-1,n-2) \leq 2n-4$.

Experimental results

Research questions

• RQ1 What is the minimum number of edges required for a connected graph of order $n$ to have $rx_3(G) \leq \ell$ for small $\ell$, such as $\ell = 2,3$?
• RQ2 Can tight upper bounds be established for $t(n,3,\ell)$ when $\ell$ ranges from $3$ to $n-3$, particularly for $\ell \geq n/2$?
• RQ3 How do graph structures like complete bipartite graphs and rose graphs influence the $3$-rainbow index, and can they be used to construct extremal examples?
• RQ4 What is the behavior of $t(n,3,\ell)$ as $\ell$ approaches $n-1$, and how does it relate to trees and unicyclic graphs with girth 3?
• RQ5 Can the $k$-rainbow index be bounded uniformly for larger $k$, such as $k=n-1$, and what is the minimal edge count in such cases?

Key findings

• For $n=3$, $t(n,3,2) = 2$; for $n=4$, $t(n,3,2) = 4$; and for $n=5$, $t(n,3,2) = \binom{5}{2} = 10$, but no such graph exists for $n \geq 6$ with $rx_3(G) \leq 2$.
• When $n \geq 6$, $t(n,3,3) \leq \lfloor n^2/4 \rfloor$ for even $n$, and $t(n,3,3) \leq \frac{(n+3)(n-1)}{4}$ for odd $n$, achieved via $K_{n/2,n/2}$ and a modified join construction.
• For $\ell$ satisfying $n/2 \leq \ell \leq n-3$, the bound $t(n,3,\ell) \leq 2n - \ell - 1$ holds, constructed by joining a $(n-\ell,3)$-rose graph to a path of length $2\ell - n$.
• The exact value $t(n,3,n-2) = n$ is established, achieved by a cycle of length $n$ with a chord or a unicyclic graph of girth 3.
• The value $t(n,3,n-1) = n-1$ is confirmed, corresponding to trees or unicyclic graphs with girth 3, which have $rx_3(G) = n-1$.
• For $k=n-1$, the bound $t(n,n-1,n-2) \leq 2n-4$ is proven using $K_{2,n-2}$ with a color assignment ensuring all $(n-1)$-subsets admit a rainbow $S$-tree.