Haj\'os conjectured that every graph containing no subdivision of the complete graph $K_{s+1}$ is properly $s$-colorable. This result was disproved by Catlin. Indeed, the maximum chromatic number of such graphs is $\Omega(s^2/\log s)$. In this paper we prove that $O(s)$ colors are enough for a weakening of this conjecture that only requires every monochromatic component to have bounded size (so-called \emph{clustered} coloring). Our approach in this paper leads to more results. Say that a graph is an {\it almost $(\leq 1)$-subdivision} of a graph $H$ if it can be obtained from $H$ by subdividing edges, where at most one edge is subdivided more than once. We prove the following (where $s \geq 2$): \begin{enumerate} \item Graphs of bounded treewidth and with no almost $(\leq 1)$-subdivision of $K_{s+1}$ are $s$-choosable with bounded clustering. \item For every graph $H$, graphs with no $H$-minor and no almost $(\leq 1)$-subdivision of $K_{s+1}$ are $(s+1)$-colorable with bounded clustering. \item For every graph $H$ of maximum degree at most $d$, graphs with no $H$-subdivision and no almost $(\leq 1)$-subdivision of $K_{s+1}$ are $\max\{s+3d-5,2\}$-colorable with bounded clustering. \item For every graph $H$ of maximum degree $d$, graphs with no $K_{s,t}$ subgraph and no $H$-subdivision are $\max\{s+3d-4,2\}$-colorable with bounded clustering. \item Graphs with no $K_{s+1}$-subdivision are $\max\{4s-5,1\}$-colorable with bounded clustering. \end{enumerate} The first result shows that the weakening of Haj\'{o}s' conjecture is true for graphs of bounded treewidth in a stronger sense; the final result is the first $O(s)$ bound on the clustered chromatic number of graphs with no $K_{s+1}$-subdivision.

中文翻译：

---

Haj\'os' 猜想的聚类变体

Haj\'os 推测每个不包含完整图 $K_{s+1}$ 的细分的图都是正确的 $s$-colorable。这一结果被卡特林否定了。实际上，此类图的最大色数是 $\Omega(s^2/\log s)$。在本文中，我们证明 $O(s)$ 颜色足以弱化这一猜想，即只要求每个单色分量都具有有界尺寸（所谓的 \emph{clustered} 着色）。我们在本文中的方法导致了更多结果。假设一个图是一个图 $H$ 的 {\it 几乎 $(\leq 1)$-subdivision}，如果它可以通过细分边从 $H$ 获得，其中最多一条边被细分多次。我们证明以下（其中 $s \geq 2$）： \begin{enumerate} \item 有界树宽的图并且几乎没有 $(\leq 1)$-$K_{s+1}$ 的细分是 $s $-choosable 有界聚类。\item 对于每个图 $H$，没有 $H$-minor 和几乎没有 $(\leq 1)$-subdivision of $K_{s+1}$ 的图是 $(s+1)$-colorable with bounded聚类。\item 对于每个最大度数最多为 $d$ 的图 $H$，没有 $H$-subdivision 和几乎没有 $(\leq 1)$-subdivision of $K_{s+1}$ 的图是 $\max \{s+3d-5,2\}$-colorable 有界聚类。\item 对于每个最大度数为 $d$ 的图 $H$，没有 $K_{s,t}$ 子图和 $H$-subdivision 的图为 $\max\{s+3d-4,2\}$ - 可着色的有界聚类。\item 没有 $K_{s+1}$-subdivision 的图是 $\max\{4s-5,1\}$-colorable 有界聚类。\end{enumerate} 第一个结果表明 Haj\'{o}s' 猜想的弱化对于有界树宽的图在更强的意义上是正确的；