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- Title
Ramsey numbers of the quadrilateral versus books.
- Authors
Li, Tianyu; Lin, Qizhong; Peng, Xing
- Abstract
A book Bn ${B}_{n}$ is a graph which consists of n $n$ triangles sharing a common edge. In this paper, we study Ramsey numbers of quadrilateral versus books. Previous results give the exact value of r(C4,Bn) $r({C}_{4},{B}_{n})$ for 1≤n≤14 $1\le n\le 14$. We aim to determine the exact value of r(C4,Bn) $r({C}_{4},{B}_{n})$ for infinitely many n $n$. To achieve this, we first prove that r(C4,B(m−1)2+(t−2))≤m2+t $r({C}_{4},{B}_{{(m-1)}^{2}+(t-2)})\le {m}^{2}+t$ for m≥4 $m\ge 4$ and 0≤t≤m−1 $0\le t\le m-1$. This improves upon a result by Faudree, Rousseau, and Sheehan which states that r(C4,Bn)≤g(g(n)), whereg(n)=n+⌊n−1⌋+2. $r({C}_{4},{B}_{n})\le g(g(n)),\hspace{0.17em}\hspace{0.17em}\text{where}\,\,g(n)=n+\lfloor \sqrt{n-1}\rfloor +2.$ Combining the new upper bound and constructions of C4 ${C}_{4}$‐free graphs, we are able to determine the exact value of r(C4,Bn) $r({C}_{4},{B}_{n})$ for infinitely many n $n$. As a special case, we show r(C4,Bq2−q−2)=q2+q−1 $r({C}_{4},{B}_{{q}^{2}-q-2})={q}^{2}+q-1$ for all prime powers q≥4 $q\ge 4$.
- Subjects
ROUSSEAU, Jean-Jacques, 1712-1778; QUADRILATERALS; RAMSEY numbers; TRIANGLES; NUMBER theory
- Publication
Journal of Graph Theory, 2023, Vol 103, Issue 2, p309
- ISSN
0364-9024
- Publication type
Article
- DOI
10.1002/jgt.22919