图的条件和唯一着色（修订后的重新提交）
Conditional and Unique Coloring of Graphs (revised resubmission)
摘要：
对于整数$k>0$和$0<r \leq Δ$（其中$r \leq k$），图$G$的条件$(k,r)$ -着色是$G$的顶点的真$k$ -着色，使得$G$中度为$d(v)$的每个顶点$v$与至少具有$\min\{r, d(v)\}$不同着色邻居的顶点相邻。图$G$具有条件$(k,r)$ -染色的最小整数$k$称为$r$阶条件色数，记为$χ_r(G)$ .对于不同的$r$的值，我们首先给出的结果（确切的值或界的$χ_r(G)$依赖于$r$）有关的图的条件着色。然后我们得到了某些参数化图的$χ_r(G)$，风车图、风车图的线图、友谊图的中图、圈的中图、友谊图的线图、完全$k$ -部图的中图、二部图的中图和齿轮图。最后介绍了\emph{unique conditional colorability}并给出了相关结果。
英文摘要：
For integers $k>0$ and $0<r \leq Δ$ (where $r \leq k$), a conditional $(k,r)$-coloring of a graph $G$ is a proper $k$-coloring of the vertices of $G$ such that every vertex $v$ of degree $d(v)$ in $G$ is adjacent to vertices with at least $\min\{r, d(v)\}$ differently colored neighbors. The smallest integer $k$ for which a graph $G$ has a conditional $(k,r)$-coloring is called the $r$th order conditional chromatic number, denoted by $χ_r(G)$. For different values of $r$ we first give results (exact values or bounds for $χ_r(G)$ depending on $r$) related to the conditional coloring of graphs. Then we obtain $χ_r(G)$ of certain parameterized graphs viz., windmill graph, line graph of windmill graph, middle graph of friendship graph, middle graph of a cycle, line graph of friendship graph, middle graph of complete $k$-partite graph, middle graph of a bipartite graph and gear graph. Finally we introduce \emph{unique conditional colorability} and give some related results.
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。
