Ramsey and Gallai-Ramsey numbers for linear forests and kipas

Abstract

For two graphs $G,H$, the \emph{Ramsey number} $r(G,H)$ is the minimum integer $n$ such that any red/blue edge-coloring of $K_n$ contains either a red copy of $G$ or a blue copy of $H$. For two graphs $G,H$, the \emph{Gallai-Ramsey number} $\operatorname{gr}_k(G:H)$ is defined as the minimum integer $n$ such that any $k$-edge-coloring of $K_n$ must contain either a rainbow copy of $G$ or a monochromatic copy of $H$. In this paper, the classical Ramsey numbers of linear forest versus kipas are obtained. We obtain the exact values of $\operatorname{gr}_k(G:H)$, where $H$ is either a path or a kipas and $G\in\{K_{1,3},P_4^+,P_5\}$ and $P_4^+$ is the graph consisting of $P_4$ with one extra edge incident with inner vertex.

Figures (4)

• Figure 1: The path $P^*$.
• Figure 2: Construct a monochromatic $P_n$.
• Figure 3: The subsets $U'$ and $Y$ of $B$.
• Figure 4: The construction of $P$.

Theorems & Definitions (65)

• Definition 1
• Definition 2
• Theorem 1.1
• Theorem 1.2
• Definition 3
• Definition 4
• Conjecture 1
• Theorem 1.3
• Remark 1.1
• Theorem 1.4: GLST87