$\triangle$-凸性中Carathéodory数与交换数的紧界
Tight bounds on the Carathéodory and exchange numbers in $\triangle$-convexity
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中文总结 AI 辅助
本文针对三角形凸性空间，指出并修复了Anand等人关于Carathéodory数与交换数上界证明中的漏洞，给出了完整证明、极值图刻画，并精确计算了块图和k-树的交换数。
中文摘要 AI 辅助
在有限简单图$G = (V,E)$上的$\triangle$-凸性空间是子集$S \subseteq V(G)$的集合$\mathcal{C}$，使得只要$x \in V(G)$与$S$中的两个顶点构成三角形，就有$x \in S$。$\mathcal{C}$的成员称为凸集，集合$S \subseteq V(G)$的凸包，记作$\operatorname{Hull}(S)$，是包含$S$的最小凸集。Carathéodory数（相应地，交换数）$c_{\triangle}(G)$（相应地，$e_{\triangle}(G)$）是$V(G)$中最大的Carathéodory（相应地，交换）独立子集的大小。Anand等人（JCMCC 126, 2025, 11--27）证明了$c_{\triangle}(G) \leq t(G)+1$和$e_{\triangle}(G) \leq t(G) + 2$，其中$t(G)$是$G$中三角形的数量，并且这些界是紧的。他们还根据$G$中非$K_2$块的数量和排列计算了块图$G$的$c_{\triangle}(G)$和$e_{\triangle}(G)$。在本文中，我们指出Anand等人关于第一个不等式$c_{\triangle}(G) \leq t(G)+1$的证明中存在一个漏洞，该漏洞也影响了第二个不等式$e_{\triangle}(G) \leq t(G) + 2$的证明。此外，紧性结果被无意中用作极值图的刻画，导致在某些情况下块图的Carathéodory数和交换数的计算错误。我们通过给出第一个不等式的完整证明来修复这些漏洞，该证明采用了与Anand等人不同的路径。结合Anand等人的论证，这也完成了第二个不等式的证明。我们的证明还引出了每个界的极值图的刻画，我们利用该刻画来计算块图的Carathéodory数和交换数，并识别极值块图。我们还确定了每个$k \geq 2$的$k$-树的$e_{\triangle}(G)$的精确值。
英文摘要
The $\triangle$-convexity space on a finite, simple graph $G = (V,E)$ is the collection $\mathcal{C}$ of subsets $S \subseteq V(G)$ such that whenever $x \in V(G)$ forms a triangle with two vertices in $S$, we have $x \in S$. The members of $\mathcal{C}$ are called convex sets, and the convex hull of a set $S \subseteq V(G)$, denoted $\operatorname{Hull}(S)$, is the smallest member of $\mathcal{C}$ that contains $S$. The Carathéodory (resp., exchange) number, $c_{\triangle}(G)$ (resp., $e_{\triangle}(G)$), is the size of a largest Carathéodory (resp., exchange) independent subset of $V(G)$. It was shown by Anand et al. (JCMCC 126, 2025, 11--27) that $c_{\triangle}(G) \leq t(G)+1$ and $e_{\triangle}(G) \leq t(G) + 2$, where $t(G)$ is the number of triangles in $G$, and that these bounds are tight. They also computed $c_{\triangle}(G)$ and $e_{\triangle}(G)$ for a block graph $G$ in terms of the number and arrangement of non-$K_2$ blocks in $G$. In this paper, we point out a gap in the proof in Anand et al. of the first inequality, $c_{\triangle}(G) \leq t(G)+1$, which has consequences for the proof of the second inequality, $e_{\triangle}(G) \leq t(G) + 2$, as well. Moreover, the tightness results are inadvertently applied as characterizations of the extremal graphs, leading to incorrect computations of the Carathéodory and exchange numbers of block graphs in certain cases. We fix these gaps by giving a full proof of the first inequality via a different route from that in Anand et al. Together with the argument in Anand et al., this also completes the proof of the second inequality. Our proof also leads to a characterization of the extremal graphs for each bound, which we use to compute the Carathéodory and exchange numbers of block graphs and to identify the extremal block graphs. We also determine $e_{\triangle}(G)$ exactly for $k$-trees for every $k \geq 2$.
发表机构
- Indian Institute of Technology Jodhpur（印度理工学院乔德普尔分校）
机构由 AI 辅助整理，请以论文原文为准。
