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Title:
Gallai-Ramsey numbers for rainbow $S^{+}_{3}$ and monochromatic paths
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Discussiones Mathematicae Graph Theory 42(2) (2022) 349-362
Received: 2018-06-13 , Revised: 2019-03-18 , Accepted: 2019-10-22 , Available online: 2020-03-04 , https://doi.org/10.7151/dmgt.2310
Abstract:
Motivated by Ramsey theory and other rainbow-coloring-related problems, we
consider edge-colorings of complete graphs without rainbow copy of some fixed
subgraphs. Given two graphs $G$ and $H$, the $k$-colored Gallai-Ramsey number
$gr_k(G : H)$ is defined to be the minimum positive integer $n$ such that every
$k$-coloring of the complete graph on $n$ vertices contains either a rainbow
copy of $G$ or a monochromatic copy of $H$. Let $S^{+}_{3}$ be the graph on
four vertices consisting of a triangle with a pendant edge. In this paper, we
prove that $gr_k(S^{+}_{3} : P_5)=k+4$ ($k\geq 5$), $gr_k(S^{+}_{3} : mP_2)=
(m-1)k+m+1$ ($k\geq 1$), $gr_k(S^{+}_{3} : P_3 \cup P_2)= k+4$ ($k\geq 5$)
and $gr_k(S^{+}_{3} : 2P_3)=k+5$ ($k\geq 1$).
Keywords:
Gallai-Ramsey number, rainbow coloring, monochromatic paths
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