arXiv subjects
Publications and source records attributed to Jędrzej Hodor.
A vertex coloring $φ$ of a graph $G$ is $p$-centered if for every connected subgraph $H$ of $G$, either $φ$ uses more than $p$ colors on $H$, or there is a color that appears exactly once on $H$. We prove that for every fixed positive integer $t$, every $K_t$-minor-free graph admits a $p$-centered coloring using $\mathcal{O}(p^{t-1})$ colors.
We give a short proof that for every apex-forest $X$ on at least two vertices, graphs excluding $X$ as a minor have layered pathwidth at most $2|V(X)|-3$. This improves upon a result by Dujmović, Eppstein, Joret, Morin, and Wood (SIDMA, 2020). Our main tool is a structural result about graphs excluding a forest as a rooted minor, which is of independent interest. We develop similar tools for treedepth and treewidth. We discuss implications for Erdős-Pósa properties of rooted models of minors in graphs.
We study the growth rate of weak coloring numbers of graphs excluding a fixed graph as a minor. Van den Heuvel et al. (European J. of Combinatorics, 2017) showed that for a fixed graph $X$, the maximum $r$-th weak coloring number of $X$-minor-free graphs is polynomial in $r$. We determine this polynomial up to a factor of $\mathcal{O}(r \log r)$. Moreover, we tie the exponent of the polynomial to a structural property of $X$, namely, $2$-treedepth. As a result, for a fixed graph $X$ and an $X$-minor-free graph $G$, we show that $\mathrm{wcol}_r(G)= \mathcal{O}(r^{\mathrm{td}(X)-1}\mathrm{log}\ r)$, which improves on the bound $\mathrm{wcol}_r(G) = \mathcal{O}(r^{g(\mathrm{td}(X))})$ given by Dujmović et al. (SODA, 2024), where $g$ is an exponential function. In the case of planar graphs of bounded treewidth, we show that the maximum $r$-th weak coloring number is in $\mathcal{O}(r^2\mathrm{log}\ r$), which is best possible.
For every integer $n$ with $n \geq 6$, we prove that the Boolean dimension of a poset consisting of all the subsets of $\{1,\dots,n\}$ equipped with the inclusion relation is strictly less than $n$.
A graph $G$ contains a graph $H$ as an induced minor if $H$ can be obtained from $G$ after vertex deletions and edge contractions. We show that for every $k$-vertex planar graph $H$, every graph $G$ excluding $H$ as an induced minor and $K_{t,t}$ as a subgraph has treewidth at most $Δ(G)^{f(k,t)}$ where $Δ(G)$ denotes the maximum degree of $G$. Without requiring the absence of a $K_{t,t}$ subgraph, Korhonen [JCTB '23] has shown the upper bound of $k^{O(1)} 2^{Δ(G)^5}$ whose dependence in $Δ(G)$ is exponential. Our result partially answers a question of Chudnovsky [Dagstuhl seminar '23] asking whether the treewidth of graphs with $Δ(G)=O(\log{|V(G)|})$ excluding both a $k$-vertex planar graph as an induced minor and the biclique $K_{t,t}$ as a subgraph is in $O_{k,t}(\log |V(G)|)$. We confirm that the treewidth is in this case polylogarithmic in $|V(G)|$.
Tsirelson's norm $\|\cdot \|_T$ on $c_{00}$ is defined as the supremum over a certain collection of iteratively defined, monotone increasing norms $\|\cdot \|_k$. For each positive integer $n$, the value $j(n)$ is the least integer $k$ such that for all $x \in \mathbb{R}^n$ (here $\mathbb{R}^n$ is considered as a subspace of $c_{00}$), $\|x\|_T = \|x\|_k$. In 1989 Casazza and Shura asked what is the order of magnitude of $j(n)$. It is known that $j(n) \in \mathcal{O}(\sqrt{n})$. We show that this bound is tight, that is, $j(n) \in Ω(\sqrt{n})$. Moreover, we compute the tight order of magnitude for some norms being modifications of the original Tsirelson's norm.
Over the past 10 years, there has been considerable interest in exploring questions connecting dimension for posets with graph theoretic properties of their cover graphs and order diagrams, especially with the concepts of planarity and treewidth. Joret and Micek conjectured that if $P$ is a poset with a planar cover graph, then the dimension of $P$ is bounded in terms of the number of minimal elements of $P$ and the treewidth of the cover graph of $P$. We settle this conjecture in the affirmative by strengthening a recent breakthrough result [14] by Blake, Micek, and Trotter, who proved that for each poset $P$ admitting a planar cover graph and a unique minimal element we have $\mathrm{dim}(P) \leq 2 \mathrm{se}(P) + 2$, namely, we prove that $\mathrm{dim}(P) \leq 2 \mathrm{wheel}(P) + 2$.
We study reconfiguration of independent sets in interval graphs under the token sliding rule. We show that if two independent sets of size $k$ are reconfigurable in an $n$-vertex interval graph, then there is a reconfiguration sequence of length $\mathcal{O}(k\cdot n^2)$. We also provide a construction in which the shortest reconfiguration sequence is of length $Ω(k^2\cdot n)$. As a counterpart to these results, we also establish that $\textsf{Independent Set Reconfiguration}$ is PSPACE-hard on incomparability graphs, of which interval graphs are a special case.