论文标题
具有小零件的多标度股多部分图
Multithreshold multipartite graphs with small parts
论文作者
论文摘要
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.图的阈值是最小的$ k $，它是$ k $ trainshord图。贾米森（Jamison）和斯普拉格（Sprague）引入了多坐孔图，作为经典阈值图的概括。他们要求提供完整多部分图的确切阈值。最近，Chen和Hao解决了每个零件不太小的完整多部分图的问题，他们要求每个零件的尺寸$ 3 $。我们确定$ k_ {3，3，\ dots，3} $，$ k_ {4，4，\ dots，4} $的确切阈值编号及其补充$ nk_3 $，$ nk_4 $。这改善了Puleo的结果。
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.
