论文标题
无间隙图的规律性的新上限
A new upper bound for the regularity of gap-free graphs
论文作者
论文摘要
在本文中，就其最小的三角剖分而言，我们为无间隙图的边缘理想的规律提供了新的上限。令$ h_u = g \ cup f_u $是无间隙图$ g $的最小三角剖分，对于某些最大独立套装$ u $ in $ g $中。令$ \ Mathcal {c} _U $为$ 3 $均匀的$ 3 $ - $ h_u $中的所有$ 3 $ paths，其中一个来自$ f_u $的边缘和另一个边缘来自$ g $。然后，我们证明$ \ displaystyle \ reg（i（g））\ leq \ reg（i（\ c_u））$。结果，我们为无间隙图的规律性提供了一般的上限。 Furthermore, if $\mathcal{H}$ is the $3$-uniform clutter consists of the $3$-cliques in $G$ or in $F_U$, and the $3$-paths in $G$ which are not $3$-cliques in $H_U$, then $\reg(I(G))\leq 3$, provided $\mathcal{H}$ is chordal.这部分回答了Há，\ cite [问题$ 6.3 $] {H14}以及Banerjee，Beyarslan和Há，\ cite [问题$ 7.1 $] {BBH19}的部分回答。
In this article, we give a new upper bound for the regularity of edge ideals of gap-free graphs, in terms of the their minimal triangulation. Let $H_U=G\cup F_U$ be a minimal triangulation of a gap-free graph $G$, for some maximal independent set $U$ in $G$. Let $\mathcal{C}_U$ be the $3$-uniform clutter of all $3$-paths in $H_U$ which consists of one edge coming from $F_U$ and another edge coming from $G$. Then we show that $\displaystyle \reg(I(G))\leq \reg(I(\C_U))$. As a consequence, we give a general upper bound for the regularity of gap-free graphs. Furthermore, if $\mathcal{H}$ is the $3$-uniform clutter consists of the $3$-cliques in $G$ or in $F_U$, and the $3$-paths in $G$ which are not $3$-cliques in $H_U$, then $\reg(I(G))\leq 3$, provided $\mathcal{H}$ is chordal. This answers partially a question raised by Há, \cite[Problem $6.3$]{h14} and by Banerjee, Beyarslan and Há, \cite[Problem $7.1$]{bbh19}.
