{
  "id": "2404.04432",
  "version": "v1",
  "published": "2024-04-05T22:17:05.000Z",
  "updated": "2024-04-05T22:17:05.000Z",
  "title": "Size Ramsey numbers of small graphs versus fans or paths",
  "authors": [
    "Yufan Li",
    "Yanbo Zhang",
    "Yunqing Zhang"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "For two graphs $G_1$ and $G_2$, the size Ramsey number $\\hat{r}(G_1,G_2)$ is the smallest positive integer $m$ for which there exists a graph $G$ of size $m$ such that for any red-blue edge-coloring of the graph $G$, $G$ contains either a red subgraph isomorphic to $G_1$, or a blue subgraph isomorphic to $G_2$. Let $P_n$ be a path with $n$ vertices, $nK_2$ a matching with $n$ edges, and $F_n$ a graph with $n$ triangles sharing exactly one vertex. If $G_1$ is a small fixed graph and $G_2$ denotes any graph from a graph class, one can sometimes completely determine $\\hat{r}(G_1,G_2)$. Faudree and Sheehan confirmed all size Ramsey numbers of $P_3$ versus complete graphs in 1983. The next year Erd\\H{o}s and Faudree confirmed that of $2K_2$ versus complete graphs and complete bipartite graphs. We obtain three more Ramsey results of this type. For $n\\ge 3$, we prove that $\\hat{r}(P_3,F_n)=4n+4$ if $n$ is odd, and $\\hat{r}(P_3,F_n)=4n+5$ if $n$ is even. This result refutes a conjecture proposed by Baskoro et al. We also show that $\\hat{r}(2K_2,F_2)=12$ and $\\hat{r}(2K_2,F_n)=5n+3$ for $n\\ge 3$. In addition, we prove that $\\hat{r}(2K_2,nP_m)=\\min\\{nm+1, (n+1)(m-1)\\}$. This result verifies a conjecture posed by Vito and Silaban.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-05T22:17:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C55",
      "05D10"
    ],
    "keywords": [
      "size ramsey number",
      "small graphs",
      "complete graphs",
      "complete bipartite graphs",
      "blue subgraph isomorphic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}