发表机构
Kennesaw State University（肯尼索州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究 Hajós 并和顶点识别下 Nullstellensatz 证书最小次数，给出上界并构造达到界的 4-临界图族，分类保持次数的识别。
AI 中文摘要
我们研究 Bayer 的 $k$-着色方程的 Nullstellensatz 证书的最小系数次数 $N_{k,\F}(G)$，其中 $\F$ 的特征不整除 $k$。若 $J$ 是非 $k$-可着色图 $G,H$ 的 \HJ\\ 并，且 $m=\max\{N_{k,\F}(G),N_{k,\F}(H)\}$，则 $N_{k,\F}(J)\leq m+k$。当删除所选边使每个输入图 $k$-可着色时，我们还有 $N_{k,\F}(J)\geq m$；次数同余则给出 $N_{k,\F}(J)\in\{m,m+k\}$。这部分回答了 Li、Lowenstein 和 Omar 的一个问题。对于 $\F_2$ 上的三着色，我们构造了一个无限 $4$-临界的精确次数为七的族，在输入次数为四时达到该界。相反，仅通过 \HJ\\ 并从 $K_4$ 构造的每个图都有次数 $O(\log n)$ 且证书具有多项式多个项：并保持树宽至多三，平衡分隔符产生低次数证书。额外的顶点识别被排除在此障碍之外。我们分类了 $25$ 顶点基图的所有单次识别；恰好 $36$ 个保持次数七，产生 $24$ 顶点的树宽为四的 $4$-临界图。在相邻真孪生上的压缩自并防止次数损失，并给出每轮增加四个顶点的可重复规则。该规则不建立次数放大或临界性的保持。精确见证和独立验证程序伴随有限结果。
英文摘要
We study the minimum coefficient degree $N_{k,\F}(G)$ of a Nullstellensatz certificate for Bayer's $k$-coloring equations, where the characteristic of $\F$ does not divide $k$. If $J$ is a \HJ\ join of non-$k$-colorable graphs $G,H$ and $m=\max\{N_{k,\F}(G),N_{k,\F}(H)\}$, then $N_{k,\F}(J)\leq m+k$. When deletion of the selected edge makes each input $k$-colorable, we also have $N_{k,\F}(J)\geq m$; the degree congruence then gives $N_{k,\F}(J)\in\{m,m+k\}$. This partially answers a question of Li, Lowenstein, and Omar. For three-coloring over $\F_2$, we construct an infinite $4$-critical family of exact degree seven, attaining the bound at input degree four. In contrast, every graph constructed from $K_4$ solely by \HJ\ joins has degree $O(\log n)$ and a certificate with polynomially many terms: joins preserve treewidth at most three, and balanced separators yield low-degree certificates. Additional vertex identifications are excluded from this obstruction. We classify all single identifications of the $25$-vertex base graph; exactly $36$ preserve degree seven, producing $24$-vertex $4$-critical graphs of treewidth four. A compressed self-join at adjacent true twins prevents degree loss and gives a repeatable rule adding four vertices per round. The rule does not establish degree amplification or preservation of criticality. Exact witnesses and standalone verification programs accompany the finite results.
