直积中的等视野、稳定性和刺激性
Isoperimetry, Stability, and Irredundance in Direct Products
摘要：
图的直积$G_1,\ldots,G_n$是具有顶点集$V(G_1)\times\cdots\times V(G_n)$的图，其中两个顶点$(g_1,\ldots,g_n)$和$(g_1',\ldots,g_n')$相邻当且仅当对于所有$i$，$g_i$与$G_i$中的$g_i'$相邻。在Brakensiek最近工作的基础上，我们证明了完全多部图的直积的一个最优点等周不等式.应用这个不等式，我们得到了平衡完全多部图的直积中独立集的稳定性结果，证明了每一个大的独立集都必须接近于通过设置其中一个坐标为常数所确定的最大独立集.利用这些等周性和稳定性结果，我们证明了平衡完全多部图的直积的上不可约数在所有情况下都等于它的独立数，但在大多数情况下是$37$.这证明了大部分的猜想Burcroff出现作为一个加强猜想的第二作者和Iyer。我们还提出了进一步加强Burcroff猜想。
英文摘要：
The direct product of graphs $G_1,\ldots,G_n$ is the graph with vertex set $V(G_1)\times\cdots\times V(G_n)$ in which two vertices $(g_1,\ldots,g_n)$ and $(g_1',\ldots,g_n')$ are adjacent if and only if $g_i$ is adjacent to $g_i'$ in $G_i$ for all $i$. Building off of the recent work of Brakensiek, we prove an optimal vertex isoperimetric inequality for direct products of complete multipartite graphs. Applying this inequality, we derive a stability result for independent sets in direct products of balanced complete multipartite graphs, showing that every large independent set must be close to the maximal independent set determined by setting one of the coordinates to be constant. Armed with these isoperimetry and stability results, we prove that the upper irredundance number of a direct product of balanced complete multipartite graphs is equal to its independence number in all but at most $37$ cases. This proves most of a conjecture of Burcroff that arose as a strengthening of a conjecture of the second author and Iyer. We also propose a further strengthening of Burcroff's conjecture.
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