具有小部分的多阈值多部图
Multithreshold multipartite graphs with small parts
摘要：
一个图是一个$k$ -阈值图与阈值$θ_1, θ_2, \dots, θ_k$，如果我们可以分配一个实数$r_v$给每个顶点$v$，使得对于任何两个不同的顶点$u$和$v$，$uv$是一条边当且仅当不超过$r_u+r_v$的阈值的数量是奇数。一个图的阈值数是它是一个$k$ -阈值图的最小$k$。多阈值图是由Jamison和Sprague作为经典阈值图的推广而提出的。他们要求完全多部图的精确阈值数。最近，Chen和Hao解决了每个部分都不太小的完全多部图的问题，并且他们要求每个部分都有大小PDMATHAMT的情况。我们确定了$K_{3, 3, \dots, 3}$、$K_{4, 4, \dots, 4}$及其互补物$nK_3$、$nK_4$的精确阈值数。这改善了Puleo的结果。
英文摘要：
A graph is a $k$-threshold graph with thresholds $θ_1, θ_2, \dots, θ_k$ if we can assign a real number $r_v$ to each vertex $v$ such that for any two distinct vertices $u$ and $v$, $uv$ is an edge if and only if the number of thresholds not exceeding $r_u+r_v$ is odd. The threshold number of a graph is the smallest $k$ for which it is a $k$-threshold graph. Multithreshold graphs were introduced by Jamison and Sprague as a generalization of classical threshold graphs. They asked for the exact threshold numbers of complete multipartite graphs. Recently, Chen and Hao solved the problem for complete multipartite graphs where each part is not too small, and they asked for the case when each part has size $3$. We determine the exact threshold numbers of $K_{3, 3, \dots, 3}$, $K_{4, 4, \dots, 4}$ and their complements $nK_3$, $nK_4$. This improves a result of Puleo.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
