Haj\'os 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 $\Omega(s^2/\log s)$. 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. Say that a graph is an 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. Note that every graph with no $H$-subdivision does not contain an almost $(\leq 1)$-subdivision of $H$. We prove the following (where $s \geq 2$): (1) Graphs of bounded treewidth and with no almost $(\leq 1)$-subdivision of $K_{s+1}$ are $s$-choosable with bounded clustering. (2) For every graph $H$, graphs with no $H$-minor and no almost $(\leq 1)$-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 $(\leq 1)$-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. 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.

翻译：Haj\dos 猜测每个含有不包含完整图形$K%3+1美元亚值的图表 $2 美元 3+1美元 美元, 美元是正色的 美元 4美元 。 这种猜测被Catlin 推翻了。 事实上, 这些图表的最大色谱数是 $( =2/\ logs) 美元 。 我们证明, $1 仅要求每个单色元部分有固定的大小( 所谓的分组色) 。 我们的方法导致更多的结果 。 说 图表是 $( leq 1 ) 美元 美元 。 如果能够从 $( =2) 美元中获取, 最大色数是 $( =2) 美元 美元 美元 美元 。 注意, 没有 $( =1) 美元 的每张数( =) 美元 。 我们证明( $ 2 美元 = 美元 : (1) 以 美元为平面 美元, 美元为平面 美元 美元 美元 美元 美元 美元 美元 。