{ "id": "1905.11380", "version": "v1", "published": "2019-05-25T15:35:23.000Z", "updated": "2019-05-25T15:35:23.000Z", "title": "On Star-critical (K1,n,K1,m + e) Ramsey numbers", "authors": [ "C. J. Jayawardene", "J. N. Senadheera", "K. A. S. N. Fernando", "W. C. W Navaratna" ], "comment": "8 pages, 5 figures", "categories": [ "math.CO" ], "abstract": "Let $K_n$ denote the complete graph on $n$ vertices and $G, H$ be finite graphs without loops or multiple edges. If for every red/blue colouring of edges of the complete graph $K_n$, there exists a red copy of $G$, or a blue copy of $H$, we will say that $K_n\\rightarrow (G,H)$. The Ramsey number $r(G, H)$ is the smallest positive integer $n$ such that $K_{n} \\rightarrow (G, H)$. Star-critical Ramsey number $r_*(G, H)$ is defined as the largest value of $k$ such that $K_{r(G,H)-1} \\sqcup K_{1,k} \\rightarrow (G, H)$. In this paper, we will find $r_*(K_{1,n}, K_{1,m}+e)$ for all $n,m \\geq 3$.", "revisions": [ { "version": "v1", "updated": "2019-05-25T15:35:23.000Z" } ], "analyses": { "keywords": [ "complete graph", "multiple edges", "blue copy", "finite graphs", "smallest positive integer" ], "note": { "typesetting": "TeX", "pages": 8, "language": "en", "license": "arXiv", "status": "editable" } } }