[Paper Review] Rainbow connection of bridgeless outerplanar graphs with small diameters

Xingchao Deng, He Song+2 more|arXiv (Cornell University)|Sep 8, 2016

Advanced Graph Theory Research10 references3 citations

TL;DR

This paper establishes exact rainbow connection numbers for bridgeless outerplanar graphs with diameter 2 or 3. It proves that diameter-2 graphs have rainbow connection number 2 except for fan graphs $F_n$ ($n \geq 7$) and $C_5$, which require 3 colors; for diameter-3 graphs, the rainbow connection number is at most 4, and this bound is tight.

ABSTRACT

In this paper, we investigate rainbow connection number $rc(G)$ of bridgeless outerplanar graphs $G$ with diameter 2 or 3. We proved the following results: If $G$ has diameter $2,$ then $rc(G)=3$ for fan graphs $F_{n}$ with $n\geq 7$ or $C_5,$ otherwise $rc(G)=2;$ if $G$ has diameter $3,$ then $rc(G)\leq 4$ and the bound is sharp.

Motivation & Objective

• To determine the rainbow connection number $rc(G)$ for bridgeless outerplanar graphs with diameter 2.
• To establish an upper bound for $rc(G)$ in bridgeless outerplanar graphs with diameter 3.
• To prove the tightness of the upper bound for diameter-3 graphs.
• To characterize the rainbow connection behavior of special outerplanar graphs such as fans and cycles.
• To provide a complete classification of rainbow connection numbers based on graph structure and diameter.

Proposed method

• Analyzing the structure of bridgeless outerplanar graphs by classifying them based on their longest induced cycles: $C_3$, $C_4$, $C_5$, or larger.
• Using inductive construction by adding vertices to base cycles ($C_4$, $C_5$) while preserving outerplanarity and bridgelessness.
• Applying rainbow coloring techniques to fan structures ($F_n$) and cycle-plus-fan configurations, ensuring edge-disjoint rainbow paths between all vertex pairs.
• Constructing explicit 4-colorings for diameter-3 graphs and proving that 3 colors are insufficient for certain extremal examples.
• Using graph minors and structural properties (e.g., maximal outerplanar graphs, connected dominating sets) to bound the rainbow connection number.
• Validating tightness by exhibiting specific graphs (e.g., in Figure 16) that require exactly 4 colors and cannot be rainbow-connected with 3.

Experimental results

Research questions

• RQ1 What is the rainbow connection number for bridgeless outerplanar graphs of diameter 2?
• RQ2 For which specific graphs (e.g., $F_n$, $C_5$) does the rainbow connection number exceed 2 in diameter-2 outerplanar graphs?
• RQ3 What is the tightest possible upper bound for the rainbow connection number of bridgeless outerplanar graphs with diameter 3?
• RQ4 Can the upper bound of 4 for diameter-3 graphs be achieved, and if so, which graphs require exactly 4 colors?
• RQ5 How do fan structures and cycle configurations influence the rainbow connection number in outerplanar graphs?

Key findings

• For bridgeless outerplanar graphs of diameter 2, the rainbow connection number is 2 except for $F_n$ with $n \geq 7$ and $C_5$, which require 3 colors.
• $rc(G) = 3$ for $F_n$ when $n \geq 7$ and for $C_5$, confirming that these graphs are the only diameter-2 outerplanar graphs requiring 3 colors.
• For bridgeless outerplanar graphs of diameter 3, $rc(G) \leq 4$, and this bound is tight, as demonstrated by explicit constructions requiring exactly 4 colors.
• The graph in Figure 16(1) has rainbow connection number 4, and its induced subgraph (Figure 16(4)) also requires 4 colors, proving tightness of the bound.
• All graphs in the family of diameter-3 outerplanar graphs can be rainbow-colored using at most 4 colors, regardless of size or fan structure complexity.
• The presence of $C_5$ or large fan structures ($F_n$, $n \geq 7$) significantly increases the rainbow connection number, while smaller cycles and fans allow 2 or 3 colors.