Hajós conjectured that every graph containing no subdivision of the complete graph $K_{s+1}$ is properly s-colorable. This conjecture was disproved by Catlin. Indeed, the maximum chromatic number of such graphs is $\mathrm{\Omega }(s^2/\log s)$. 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 clustered coloring).

Our approach leads to more results, many of which only require a much weaker assumption that forbids an ‘almost $(⩽1)$-subdivision’ (where at most one edge is subdivided more than once). This assumption is best possible, since no bound on the number of colors exists unless we allow at least one edge to be subdivided arbitrarily many times. We prove the following (where $s⩾2$):

1.

Graphs of bounded treewidth and with no almost $(⩽1)$-subdivision of $K_{s+1}$ are s-choosable with bounded clustering.

2.

For every graph H, graphs with no H-minor and no almost $(⩽1)$-subdivision of $K_{s+1}$ are $(s+1)$-colorable with bounded clustering.

3.

For every graph H of maximum degree at most d, graphs with no H-subdivision and no almost $(⩽1)$-subdivision of $K_{s+1}$ are $\max \{s+3d-5,2\}$-colorable with bounded clustering.

4.

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.

5.

Graphs with no $K_{s+1}$-subdivision are $(4s-5)$-colorable with bounded clustering.

Hajós 猜想每个图都不包含完整图的细分 ${钾}_{秒+1}$是正确的可着色。这一猜想被卡特林推翻。事实上，这些图的最大色数是$\mathrm{\Omega }({秒}^2/\mathrm{日志}秒)$. 我们证明$哦(秒)$颜色足以削弱这一猜想，即只要求每个单色分量都具有有界尺寸（所谓的聚类着色）。

我们的方法导致了更多的结果，其中许多只需要一个更弱的假设，即禁止“几乎 $(⩽1)$-subdivision'（最多将一条边细分不止一次）。这种假设是最好的可能，因为除非我们允许至少一条边被任意细分多次，否则不存在颜色数量的限制。我们证明以下（其中$秒⩾2$)：

1.

有界树宽图且几乎没有 $(⩽1)$-细分 ${钾}_{秒+1}$是s -choosable 有界聚类。

2.

对于每个图H，没有H -minor 和几乎没有的图$(⩽1)$-细分 ${钾}_{秒+1}$ 是 $(秒+1)$- 可着色的有界聚类。

3.

对于最大度数最多为d 的每个图H，没有H细分且几乎没有H细分的图$(⩽1)$-细分 ${钾}_{秒+1}$ 是 $\mathrm{最大限度}\{秒+3d-5,2\}$- 可着色的有界聚类。

4.

对于最大度数d 的每个图H，没有${钾}_{秒,吨}$子图和没有H细分是$\mathrm{最大限度}\{秒+3d-4,2\}$- 可着色的有界聚类。

5.

没有的图 ${钾}_{秒+1}$-细分是 $(4秒-5)$- 可着色的有界聚类。