@book{844a72ca8db64636a548fa56dcbcaafc,

title = "Path-fan Ramsey numbers",

abstract = "For two given graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest positive integer $p$ such that for every graph $F$ on $p$ vertices the following holds: either $F$ contains $G$ as a subgraph or the complement of $F$ contains $H$ as a subgraph. In this paper, we study the Ramsey numbers $R(P_{n},F_{m})$, where $P_{n}$ is a path on $n$ vertices and $F_{m}$ is the graph obtained from $m$ disjoint triangles by identifying precisely one vertex of every triangle ($F_{m}$ is the join of $K_{1}$ and $mK_{2}$). We determine exact values for $R(P_{n},F_{m})$ for the following values of $n$ and $m$: $n=1,2$ or $3$ and $m\geq 2$; $n\geq 4$ and $2\leq m\leq (n+1)/2$; $n\geq7$ and $m=n-1$ or $m=n$; $n\geq 8$ and $(k\cdot n-2k+1)/2\leq m\leq (k\cdot n-k+2)/2$ with $3\leq k\leq n-5$; $n=4,5$ or $6$ and $m\geq n-1$; $n\geq 7$ and $m\geq (n-3)^2/2$. We conjecture that $R(P_{n},F_{m}) \leq 2m+n-3$ for the other values of $m$ and $n$.",

keywords = "MSC-05C55, EWI-3523, IR-65888, MSC-05D10",

author = "M. Salman and Broersma, {Haitze J.}",

note = "Imported from MEMORANDA",

year = "2003",

language = "English",

series = "Memorandum / Department of Applied Mathematics",

publisher = "University of Twente, Department of Applied Mathematics",

number = "1703",

}