通过加权局部覆盖求多重集轮廓图中的独立集
Independent Sets in Multiset Profile Graphs via Weighted Local Covers
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中文总结 AI 辅助
研究多重集轮廓图中独立集最大规模\(\alpha_q(d)\)，通过加权局部覆盖方法，给出\(q = 3\)、\(q = 4\)新证明及\(q = 5\)、\(q = 7\)精确值，还解决\(d\)固定\(q\)增长时多度数问题并改进渐近上界。
中文摘要 AI 辅助
离散单纯形由坐标和为\(d\)的非负整数向量\(a=(a_1,\ldots,a_q)\)组成，其顶点是\(q\)个符号上大小为\(d\)的多重集的多重性轮廓。两个顶点相邻当一个通过将一个坐标减\(1\)并将另一个坐标加\(1\)从另一个得到。我们研究此图中独立集的最大规模\(\alpha_q(d)\)。我们的上界通过平移较小的图覆盖该图并为它们分配非负权重。对于固定的\(q\)，权重仅取决于有限多个有界轮廓，所以一个有限有理线性系统可以为每个足够大的\(d\)证明一个界。该方法给出了\(q = 3\)和\(q = 4\)已知情况的新证明，并精确确定了\(q = 5\)和\(q = 7\)每个度数下的\(\alpha_q(d)\)。它还确定了一般\(q\)的最大自然加性着色类。在相反的情况下，\(d\)固定且\(q\)增长，它解决了\(q\geq7\)时的五度问题，给出了六度、八度和十度的精确二的幂次族，并为每个固定的\(d\geq7\)通过三项给出了渐近尖锐的上界。最后这个结果改进了先前已知的渐近上界。
英文摘要
Let $G_q(d)$ be the unit-transfer graph on the nonnegative integer vectors whose $q$ coordinates sum to $d$, equivalently on the multiplicity profiles of size-$d$ multisets over $q$ symbols. The prime-checksum conjecture predicts that, for prime $q$ and all sufficiently large $d$, a largest independent set is a fiber of the natural cyclic checksum. We develop a finite-state weighted local-cover method for $G_q(d)$: translated induced subgraphs give local independence inequalities, while capped anchor profiles reduce the covering conditions for infinitely many degrees to a finite rational linear system. This method gives new proofs of the known cases $q=3$ and $q=4$ and determines $α(G_q(d))$ exactly for $q=5$ and $q=7$ in every degree, thereby proving the next two odd-prime cases of the conjecture. In the complementary regime where $d$ is fixed and $q$ grows, the same method gives an explicit upper bound for $α(G_q(5))$ for every $q\ge7$, determines $α(G_q(d))$ exactly when $q$ is a power of two and $d\in\{6,8,10\}$, and yields an asymptotically sharp upper bound through three terms for every fixed $d\ge7$. The finite systems arising in these arguments are verified in exact arithmetic and supported by independently checkable certificates.
