Ramsey numbers of the quadrilateral versus books
Ramsey numbers of the quadrilateral versus books
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DOI：
10.1002/jgt.22919
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发表时间：
2022-12
影响因子：
0.9
通讯作者：
Tianyu Li;Qizhong Lin;Xing Peng
中科院分区：
文献类型：
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作者：
Tianyu Li;Qizhong Lin;Xing Peng
Bn ${B}_{n}$是一个由n个$n$三角形组成的图，它们共用一条边。本文研究了四边形与书本的拉姆齐数。先前的结果给出了r(C4,Bn) $r({C}_{4},{B}_{n})$对于1≤n≤14 $1\le n\le 14$的精确值。我们的目标是确定无穷多个n $n$的r(C4,Bn) $r({C}_{4},{B}_{n})$的确切值。为了实现这一点，我们首先证明了r(C4，B(m−1)2+(t−2))≤m2+t $r({C}_{4},{B}_{{(m-1)}^{2}+(t-2)})\le {m}^{2}+t$对于m≥4 $m\ge 4$和0≤t≤m−1 $0\le t\le m-1$。这改进了Faudree， Rousseau和Sheehan的结论，即r(C4,Bn)≤g(g(n))，其中(n)=n+⌊n−1⌋+2。$r({C}_{4},{B}_{n})\le g(g(n)),\hspace{0.17em}\hspace{0.17em}\text{where}\,\,g(n)=n+\lfloor \sqrt{n-1}\rfloor +2.$结合新的上界和C4 ${C}_{4}$ - free图的构造，我们能够确定无穷多个n的精确值r(C4,Bn) $r({C}_{4},{B}_{n})$$n$。作为一种特殊情况，我们证明r(C4,Bq2−q−2)=q2+q−1 $r({C}_{4},{B}_{{q}^{2}-q-2})={q}^{2}+q-1$对于所有素数幂q≥4 $q\ge 4$。
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 $q\ge 4$ .