Rainbow Connection for Complete Multipartite Graphs

TL;DR

The paper determines the exact threshold function $f(k,t)$ for complete multipartite graphs, proving $f(k,t)=\left\lceil \frac{2k}{t-1} \right\rceil$ for all $k,t\ge 2$. It achieves this via explicit upper-bound colorings: a $4$-coloring for bipartite graphs and a $3$-coloring construction $c_{t,K}$ for $t\ge 3$, ensuring rainbow $k$-connectivity with $\ell$ colors. A matching lower bound is established through pigeonhole arguments, showing that smaller part sizes cannot guarantee such connectivity, hence $f(k,t)$ cannot be lower. The authors further identify when $\mathrm{rc}_2(K_{n_1,\dots,n_t})=2$, presenting 2-color constructions (including a bit-string approach) for graphs like $K_{m,n,n}$ and a concrete example $K_{2,4,16}$. Together, these results resolve the Fujita–Liu–Magnant question and deepen understanding of rainbow $k$-connectivity in unbalanced complete 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 $k$-connected if every pair of vertices is connected by $k$ internally disjoint rainbow paths. The rainbow $k$-connection number $\mathrm{rc}_k(G)$ is the minimum number of colors $\ell$ such that there exists a coloring with $\ell$ 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 $\mathrm{rc}_k(G) \le 4$ if $t=2$ and $\mathrm{rc}_k(G) \le 3$ if $t \ge 3$. Answering a question of Fujita, Liu and Magnant, we show that \[ f(k,t) = \left\lceil \frac{2k}{t-1} \right\rceil \] for all $k\geq 2$, $t\geq 2$. We also give some conditions for which $\mathrm{rc}_k(G) \le 3$ if $t=2$ and $\mathrm{rc}_k(G) \le 2$ if $t \ge 3$.

Figures (4)

• Figure 1: A rainbow $k$-connected coloring of $K_{a,b}$ when $a,b\geq 2k$.
• Figure 2: A $9$-partite graph $K$ with edges colored according to $c_{9,K}$
• Figure 3: Three pairwise internally disjoint rainbow $u,v$-paths on a 4-partite graph $K$ colored according to $c_{4,K}$.
• Figure 4: The coloring of $K_{2,4,16}$ used in the proof of Theorem \ref{['thm: 2-4-16']}. Here $C_L$ and $C_R$ represent the sets $\{c_1, \ldots, c_8\}$ and $\{c_1', \ldots, c_8'\}$, respectively.

Theorems & Definitions (18)

• Definition 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 3.1
• proof