最大化具有禁止性质的 $K_{r+1}$-自由图中的团数
Maximizing the number of cliques in $K_{r+1}$-free graphs with forbidden properties
浏览论文内容
中文总结 AI 辅助
本文研究$K_{r+1}$-自由图中避免若干哈密顿相关性质时的最大边数和$t$-团数，推广了先前结果，并确定了所有极值图。
中文摘要 AI 辅助
Ferrero和Lesniak在2018年找到了非哈密顿的$r$-部图中的最大边数。最近，我们找到了$K_{r+1}$-自由图中（1）非哈密顿的或（2）满足与Pósa定理相关的低度顶点条件的图的最大边数和$t$-团数。应用定理（2），我们在此将定理（1）从哈密顿性推广到其他性质。我们确定了避免以下性质之一的$K_{r+1}$-自由图中的最大边数和$t$-团数：可迹性、哈密顿连通性、$k$-路径哈密顿性、$k$-哈密顿性、$k$-哈密顿连通性和$k$-连通性。我们找到了所有具有最大边数的极值图。在此过程中，我们证明了避免任意一个对足够大的完全图成立的稳定性质的$K_{r+1}$-自由图的边数和$t$-团数的上界。
英文摘要
Ferrero and Lesniak in 2018 found the maximum numbers of edges in $r$-partite non-Hamiltonian graphs. Recently we found the maximum numbers of edges and $t$-cliques in $K_{r+1}$-free graphs (1) that are not Hamiltonian or (2) that satisfy a condition on low-degree vertices related to Pósa's theorem. Applying theorem (2), here we extend theorem (1) from Hamiltonicity to other properties. We determine the maximum numbers of edges and $t$-cliques in $K_{r+1}$-free graphs that avoid one of the following properties: traceability, Hamiltonian-connectedness, $k$-path Hamiltonicity, $k$-Hamiltonicity, $k$-Hamiltonian-connectedness, and $k$-connectedness. We find all extremal graphs having the maximum numbers of edges. On the way, we prove upper bounds on the numbers of edges and $t$-cliques in $K_{r+1}$-free graphs that avoid an arbitrary stable property that holds for sufficiently large complete graphs.
