Rainbow Connection for Complete Multipartite Graphs
Igor Araujo, Kareem Benaissa, Richard Bi, Sean English, Shengan Wu, Pai Zheng

TL;DR
This paper determines the minimum size of parts in complete multipartite graphs needed to guarantee low rainbow k-connection numbers, solving an open problem and providing new bounds for such graphs.
Contribution
It establishes an exact formula for the function f(k,t) related to rainbow k-connection in complete multipartite graphs, answering a question posed by Fujita, Liu, and Magnant.
Findings
f(k,t) = ceil(2k/(t-1)) for all k≥2, t≥2
Provided conditions for rainbow k-connection numbers to be at most 3 or 2
Solved an open problem in the theory of rainbow connectivity in multipartite graphs
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 -connected if every pair of vertices is connected by internally disjoint rainbow paths. The rainbow -connection number is the minimum number of colors such that there exists a coloring with colors that makes rainbow -connected. Let be the minimum integer such that every -partite graph with part sizes at least has if and if . Answering a question of Fujita, Liu and Magnant, we show that \[ f(k,t) = \left\lceil \frac{2k}{t-1} \right\rceil \] for all , . We also give some conditions for which if and if .
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