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arXiv · 1908.02348

Ramsey and Gallai-Ramsey numbers for stars with extra independent edges

Yaping Mao·Zhao Wang·Colton Magnant·Ingo Schiermeyer
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Abstract

Given a graph $G$ and a positive integer $k$, define the \emph{Gallai-Ramsey number} to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $G$. In this paper, we obtain general upper and lower bounds on the Gallai-Ramsey numbers for the graph $G = S_t^{r}$ obtained from a star of order $t$ by adding $r$ extra independent edges between leaves of the star so there are $r$ triangles and $t - 2r - 1$ pendent edges in $S_t^{r}$. We also prove some sharp results when $t = 2$.

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Yaping Mao, Zhao Wang, Colton Magnant, Ingo Schiermeyer. 2019-08-06. Ramsey and Gallai-Ramsey numbers for stars with extra independent edges. https://arxiv.org/abs/1908.02348

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