类 Sierpiński 图与 Kneser 图的单射着色
Injective colorings of Sierpiński-like graphs and Kneser graphs
AI总结：
本文建立单射色数与色数、边色数的关系，确定 Sierpiński 图及根积的单射色数，并刻画 Sierpiński 图与 Kneser 图的完美单射可着色性。
摘要：
本文证明了单射色数分别与色数和边色数之间的两种关系。这些关系被用于确定 Sierpiński 图的单射色数，并给出 Sierpiński 图属于第 1 类图的简短证明。文中还考虑了类 Sierpiński 图，包括环上的广义 Sierpiński 图和根积。证明了两个图的根积的单射色数处于一个包含六种可能值的集合中。文章从可完美单射着色的角度考察了 Sierpiński 图和 Kneser 图 K(n,r)；若一个图存在一种单射着色，且其中每个颜色类都构成具有最大基数的开装填，则称该图可完美单射着色。特别地，所有 Sierpiński 图以及满足 n≥3r−1 的 Kneser 图 K(n,r) 都是可完美单射着色图，而 K(7,3) 则不是。
英文摘要：
Two relationships between the injective chromatic number and, respectively, chromatic number and chromatic index, are proved. They are applied to determine the injective chromatic number of Sierpiński graphs and to give a short proof that Sierpiński graphs are Class $1$. Sierpiński-like graphs are also considered, including generalized Sierpiński graphs over cycles and rooted products. It is proved that the injective chromatic number of a rooted product of two graphs lies in a set of six possible values. Sierpiński graphs and Kneser graphs $K(n,r)$ are considered with respect of being perfect injectively colorable, where a graph is perfect injectively colorable if it has an injective coloring in which every color class forms an open packing of largest cardinality. In particular, all Sierpiński graphs and Kneser graphs $K(n, r)$ with $n \ge 3r-1$ are perfect injectively colorable graph, while $K(7,3)$ is not.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
