Tree-chromatic number is a chromatic version of treewidth, where the cost of a bag in a tree-decomposition is measured by its chromatic number rather than its size. Path-chromatic number is defined analogously. These parameters were introduced by Seymour (JCTB 2016). In this paper, we survey all the known results on tree- and path-chromatic number and then present some new results and conjectures. In particular, we propose a version of Hadwiger's Conjecture for tree-chromatic number. As evidence that our conjecture may be more tractable than Hadwiger's Conjecture, we give a short proof that every $K_5$-minor-free graph has tree-chromatic number at most $4$, which avoids the Four Colour Theorem. We also present some hardness results and conjectures for computing tree- and path-chromatic number.

中文翻译：

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树色数和路径色数的注意事项

树色数是树宽的色版本，其中树分解中包的成本是通过其色数而不是大小来衡量的。路径色数的定义类似。这些参数由 Seymour (JCTB 2016) 引入。在本文中，我们调查了树色数和路径色数的所有已知结果，然后提出了一些新的结果和猜想。特别是，我们提出了一个关于树色数的 Hadwiger 猜想的版本。作为证明我们的猜想可能比 Hadwiger 猜想更易于处理的证据，我们给出了一个简短的证明，即每个 $K_5$-minor-free 图具有最多 $4$ 的树色数，从而避免了四色定理。我们还提供了一些用于计算树色数和路径色数的硬度结果和猜想。