Rainbow Connection for Complete Multipartite Graphs
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866912715596890112 |
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| author | Araujo, Igor Benaissa, Kareem Bi, Richard English, Sean Wu, Shengan Zheng, Pai |
| author_facet | Araujo, Igor Benaissa, Kareem Bi, Richard English, Sean Wu, Shengan Zheng, Pai |
| contents | 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 $t \ge 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_12291 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Rainbow Connection for Complete Multipartite Graphs Araujo, Igor Benaissa, Kareem Bi, Richard English, Sean Wu, Shengan Zheng, Pai Combinatorics 05C15, 05C38, 05C40 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 $t \ge 3$. |
| title | Rainbow Connection for Complete Multipartite Graphs |
| topic | Combinatorics 05C15, 05C38, 05C40 |
| url | https://arxiv.org/abs/2210.12291 |