Journal of Combinatorial Optimization ( IF 1 ) Pub Date : 2023-07-24 , DOI: 10.1007/s10878-023-01068-9 Yan Li , Yusheng Li , Ye Wang
For graphs F, G and H, let \(F\rightarrow (G,H)\) signify that any red-blue edge coloring of F contains either a red G or a blue H, hence the Ramsey number R(G, H) is the smallest r such that \(K_r\rightarrow (G,H)\). Define \(K_t\) as the surplus clique of (G, H) if \(K_r\setminus K_t\rightarrow (G,H)\), where \(r=R(G,H)\). For any graph G with \(s(G)=1\), we shall show that the maximum order of surplus clique of \((G, P_n)\) is exactly \(\lceil \frac{n}{2}\rceil \) for large n.
中文翻译:
从图和路径的拉姆齐图中删除的最大集团
对于图F、G和H,让\(F\rightarrow (G,H)\)表示F的任何红蓝边着色都包含红色G或蓝色H,因此拉姆齐数R ( G , H ) 是满足\(K_r\rightarrow (G,H)\) 的最小r。如果\ (K_r\setminus K_t\rightarrow ( G , H )\) ,则将 \(K_t\)定义为 ( G , H ) 的剩余团,其中\(r=R(G,H)\)。对于任何具有\(s(G)=1\)的图G,我们将证明对于大的n ,\((G, P_n)\)的剩余团的最大阶数恰好是\(\lceil \frac{n}{2}\rceil \)。