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For positive integers $n$ and $d$, the $d$-dimensional grid of side length $n$ is the graph $G(n,d)$ with vertex set $[n]^d$, in which two vertices are adjacent if they differ by $1$ in exactly one coordinate and agree in all the other coordinates. We determine, for every fixed $d$, the asymptotic growth in $r$ of the...
We show that if a complete binary tree of height $h$ is isomorphic to a subgraph of the strong product of a graph $H$ and a path, then $\mathrm{pw}(H)$ is $\Omega(h)$. This solves a problem posed by Bose, Dujmovi\'c, Javarsineh, Morin, and Wood (2022). The proof was found by OpenAI's GPT-5.6 Sol Pro.
We prove that, for all positive integers $h$ and $t$ and every graph $X$ with $\mathrm{td}(X) \leq h$, there exists a positive integer $c(X,t)$ such that every graph $G$ with $\mathrm{tw}(G)<t$ that excludes $X$ as a topological minor is isomorphic to a subgraph of $H \boxtimes K_{c(X,t)}$ for some graph $H$ with $\mat...
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...
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