密度区域、整数证书与整数距离图的堆积着色
Density regions, integer certificates and packing colorings of distance graphs
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中文总结 AI 辅助
本文研究整数距离图的堆积着色密度区域，确定D(1,6)的最大组合密度，给出稳定索引和色数界，结合整数证书与构造证明。
中文摘要 AI 辅助
我们研究整数距离图的堆积着色中的同时颜色密度。对于 $D(1,6)$，我们确定了几个精确的密度区域，并证明颜色 $1$ 到 $7$ 的最大组合密度为 $211/252$。当接近这个最大值时，七个单独颜色的频率被迫收敛到一个指定的向量。在一个最优的低色层上，一些密度向量具有非周期实现但没有周期实现；我们确定了在相关配置之间切换所需的累积密度损失量。对于足够大的附加颜色索引，一个固定的有限图描述了联合密度区域。特别地，我们确定了每个 $i\equiv8\pmod{14}$ 且 $i\ge36$ 的七顶点区域，并证明 $36$ 是该剩余类中的第一个稳定索引。证明结合了有限状态整数证书与显式构造和极限论证。应用给出 $17\le\chi_\rho(D(1,6))\le20$，$18\le\chi_\rho(D(1,8))\le22$，以及 $\chi_\rho(D(1,9))\le17$。
英文摘要
We study simultaneous color densities in packing colorings of integer distance graphs. For $D(1,6)$, we determine several exact density regions and prove that colors $1$ through $7$ have maximum combined density $211/252$. When this maximum is approached, the seven individual color frequencies are forced to converge to a specified vector. On an optimal low-color layer, some density vectors have nonperiodic realizations but no periodic realization; we determine how much accumulated density loss is necessary for switching between the relevant configurations. For sufficiently large additional color indices, a fixed finite graph describes the joint density region. In particular, we determine a seven-vertex region for every $i\equiv8\pmod{14}$ with $i\ge36$ and prove that $36$ is the first stable index in this residue class. The proofs combine finite-state integer certificates with explicit constructions and limit arguments. Applications give $17\leχ_ρ(D(1,6))\le20$, $18\leχ_ρ(D(1,8))\le22$, and $χ_ρ(D(1,9))\le17$.
发表机构
- University of Toronto Scarborough（多伦多大学士嘉堡校区）
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