Author

Jędrzej Hodor

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Jul 2026

Optimal tree-decompositions with bags of bounded pathwidth

We show that every planar graph has a tree-decomposition with optimal width such that the subgraph induced by each bag has pathwidth at most 3. This bound is best possible, and for tree-decompositions that satisfy a certain minimality condition, we in fact give a precise description of the possible structures in each bag. Moreover, we show that the union of any $k$ bags has pathwidth $O(k)$. We also show that graphs excluding a fixed double-apex-forest minor have a tree-decomposition with optimal width such that the subgraph induced by each bag has bounded pathwidth. This includes graphs embeddable on any fixed surface. As a byproduct of our machinery, we give a new proof of the linear grid minor theorem for planar graphs.

Kevin Hendrey, Robert Hickingbotham, Jędrzej Hodor et al. · arXiv.org · 0 citations

Preprint Aug 2026

Superlinear separation between linear and centered colorings

A vertex-coloring of a graph is centered if every connected subgraph has a vertex with a unique color. A vertex-coloring of a graph is linear if every path in the graph has a vertex with a unique color. Let $\chi_{\mathrm{cen}}(G)$ and $\chi_{\mathrm{lin}}(G)$ be the minimum number of colors in a centered (resp. linear) coloring of $G$. We present a family of graphs witnessing that if $f$ is a nondecreasing function such that $\chi_{\mathrm{cen}}(G) \leq f(\chi_{\mathrm{lin}}(G))$ for every graph $G$, then $f(k) = \Omega(k^2 / \log k)$. The construction was found by OpenAI's GPT-5.6 Sol Pro.

Jędrzej Hodor, P. Micek · 0 citations