Rainbow connection for complete multipartite graphs

Araujo, Igor; Benaissa, Kareem; Bi, Richard; English, Sean; Wu, Shengan; Zheng, Pai

doi:10.2140/involve.2025.18.755

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Rainbow connection for complete multipartite graphs

Igor Araujo, Kareem Benaissa, Richard Bi, Sean English, Shengan Wu and Pai Zheng

Vol. 18 (2025), No. 5, 755–766

DOI: 10.2140/involve.2025.18.755

Abstract

A path in an edge-colored graph is said to be rainbow if no color repeats on it. An edge-colored graph is said to be rainbow $k$ -connected if there exist $k$ internally disjoint rainbow paths between each pair of vertices. The rainbow $k$ -connection number ${rc}_{k} (G)$ is the minimum number of colors $ℓ$ such that there exists a coloring with $ℓ$ colors that makes $G$ rainbow $k$ -connected. Let $f (k, t)$ be the minimum integer such that every $t$ -partite graph with part sizes at least $f (k, t)$ has ${rc}_{k} (G) \leq 4$ if $t = 2$ and ${rc}_{k} (G) \leq 3$ if $t \geq 3$ . Answering a question of Fujita, Liu and Magnant, we show that

f (k, t) = ⌈ \frac{2 k}{t - 1} ⌉

for all $k \geq 2$ , $t \geq 2$ . We also give some conditions for which ${rc}_{k} (G) \leq 3$ if $t = 2$ and ${rc}_{k} (G) \leq 2$ if $t \geq 3$ .

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Keywords

rainbow connection, multipartite

Mathematical Subject Classification

Primary: 05C15, 05C38, 05C40

Milestones

Received: 21 October 2022

Revised: 30 October 2023

Accepted: 2 July 2024

Published: 20 November 2025

Communicated by Kenneth S. Berenhaut

Authors

Igor Araujo
University of Illinois Urbana-Champaign Urbana, IL United States

Kareem Benaissa
University of Illinois Urbana-Champaign Urbana, IL United States

Richard Bi
University of Illinois Urbana-Champaign Urbana, IL United States

Sean English
University of North Carolina Wilmington Wilmington, NC United States

Shengan Wu
University of Illinois Urbana-Champaign Urbana, IL United States

Pai Zheng
University of Illinois Urbana-Champaign Urbana, IL United States

Vol. 18, No. 5, 2025