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Paul Wollan

Publications and source records attributed to Paul Wollan.

At least 19 recordsLinked to original sources
arXiv2026

An Erdős-Pósa theorem for cycles and faces of distinct lengths

We show that for every $k \in \mathbb{N}$, every graph $G$ contains $k$ vertex-disjoint cycles of different lengths, or there exists a set $X \subseteq V(G)$ with $|X| \in \mathcal{O}(k^6\mathsf{polylog}(k))$ such that $G-X$ has at most $k-1$ cycle lengths. We also prove analogous results for facial lengths of embedded graphs. Let $G$ be a graph with a closed 2-cell embedding $ψ$ on a surface $Σ$ of Euler genus $g$, let $c$ be a colouring of the faces $\mathcal{F}(ψ)$ of $ψ$, and let $R(G,ψ)$ be the radial graph of $(G, ψ)$. Then there exist $k$ faces $F_1, \ldots , F_k \in \mathcal{F}(ψ)$ that are given pairwise distinct colours by $c$ and are pairwise at distance at least $d$ in $ψ$, or there exists a set $X \subseteq V(G)$ of order at most $\mathcal{O}(k^2dg)$ such that $|\{ c(F) \mid F \in \mathcal{F}(ψ) \text{ and } V(F) \cap \bigcup_{x \in X} N^d_{R(G,ψ)}(x) = \emptyset \}| \leq k(k+2)$. Finally, using a result from additive combinatorics, we show that there are subdivided ladders with only a small number of cycle lengths. This suggests that it may be difficult to improve our bounds.

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arXiv2026

A structural duality for path-decompositions into parts of small radius

It is an easy observation that if a graph~$G$ admits a path-decomposition whose parts have small radius, then $G$ contains no large subdivision of $K_{1,3}$ or $K^3$ as a (quasi-)geodesic subgraph. We show that these are in fact the only obstructions to such path-decompositions of small radial width, and we prove analogous results for decompositions modelled on cycles and subdivided stars instead of paths. With our results we confirm in a strong form a conjecture of Georgakopoulos and Papasoglu on fat-minor-characterisations of graphs quasi-isometric to paths, cycles and paths, and subdivided stars, respectively. For this, we present a novel view on quasi-isometries between graphs by graph-decompositions of bounded radial width and spread. This new perspective enables us to prove further results in coarse graph theory, and may thus be of independent interest.

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arXiv2026

The structure of group-labeled graphs forbidding an immersion

A $Γ$-labeled graph is an oriented graph with edges invertibly labeled by a group $Γ$. We prove a structure theorem for $Γ$-labeled graphs which forbid a fixed $Γ$-labeled graph as an immersion, for any finite $Γ$. Roughly, we show that such graphs admit a tree-cut decomposition in which every bag either contains few high degree vertices or is nearly signed over a proper subgroup of $Γ$.

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arXiv2025

A characterisation of graphs quasi-isometric to $K_4$-minor-free graphs

We prove that there is a function $f$ such that every graph with no $K$-fat $K_4$ minor is $f(K)$-quasi-isometric to a graph with no $K_4$ minor. This solves the $K_4$-case of a general conjecture of Georgakopoulos and Papasoglu. Our proof technique also yields a new short proof of the respective $K_4^-$-case, which was first established by Fujiwara and Papasoglu.

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arXiv2025

Small hitting sets for longest paths and cycles

Motivated by an old question of Gallai (1966) on the intersection of longest paths in a graph and the well-known conjectures of Lovász (1969) and Thomassen (1978) on the maximum length of paths and cycles in vertex-transitive graphs, we present improved bounds for the parameters $\mathrm{lpt}(G)$ and $\mathrm{lct}(G)$, defined as the minimum size of a set of vertices in a graph $G$ hitting all longest paths (cycles, respectively). First, we show that every connected graph $G$ on $n$ vertices satisfies $\mathrm{lpt}(G)\le \sqrt{8n}$, and $\mathrm{lct}(G)\le \sqrt{8n}$ if $G$ is additionally $2$-connected. This improves a sequence of earlier bounds for these problems, with the previous state of the art being $O(n^{2/3})$. Second, we show that every connected graph $G$ satisfies $\mathrm{lpt}(G)\le O(\ell^{5/9})$, where $\ell$ denotes the maximum length of a path in $G$. As an immediate application of this latter bound, we present further progress towards Lovász' and Thomassen's conjectures: We show that every connected vertex-transitive graph of order $n$ contains a cycle (and path) of length $Ω(n^{9/14})$. This improves the previous best bound of the form $Ω(n^{13/21})$. Interestingly, our proofs make use of several concepts and results from structural graph theory, such as a result of Robertson and Seymour (1990) on transactions in societies and Tutte's $2$-separator theorem.

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arXiv2025

A grid theorem for strong immersions of walls

We show that a graph contains a large wall as a strong immersion minor if and only if the graph does not admit a tree-cut decomposition of small `width', which is measured in terms of its adhesion and the path-likeness of its torsos.

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arXiv2024

On vertex sets inducing tangles

Diestel, Hundertmark and Lemanczyk asked whether every $k$-tangle in a graph is induced by a set of vertices by majority vote. We reduce their question to graphs whose size is bounded by a function in $k$. Additionally, we show that if for any fixed $k$ this problem has a positive answer, then every $k$-tangle is induced by a vertex set whose size is bounded in $k$. More generally, we prove for all $k$ that every $k$-tangle in a graph $G$ is induced by a weight function $V(G) \to \mathbb{N}$ whose total weight is bounded in $k$. As the key step of our proofs, we show that any given $k$-tangle in a graph $G$ is the lift of a $k$-tangle in some topological minor of $G$ whose size is bounded in $k$.

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arXiv2024

A Menger-type theorem for two induced paths

We give an approximate Menger-type theorem for when a graph $G$ contains two $X-Y$ paths $P_1$ and $P_2$ such that $P_1 \cup P_2$ is an induced subgraph of $G$. More generally, we prove that there exists a function $f(d) \in O(d)$, such that for every graph $G$ and $X,Y \subseteq V(G)$, either there exist two $X-Y$ paths $P_1$ and $P_2$ such that the distance between $P_1$ and $P_2$ is at least $d$, or there exists $v \in V(G)$ such that the ball of radius $f(d)$ centered at $v$ intersects every $X-Y$ path.

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arXiv2021

Excluding a ladder

A ladder is a $2 \times k$ grid graph. When does a graph class $\mathcal{C}$ exclude some ladder as a minor? We show that this is the case if and only if all graphs $G$ in $\mathcal{C}$ admit a proper vertex coloring with a bounded number of colors such that for every $2$-connected subgraph $H$ of $G$, there is a color that appears exactly once in $H$. This type of vertex coloring is a relaxation of the notion of centered coloring, where for every connected subgraph $H$ of $G$, there must be a color that appears exactly once in $H$. The minimum number of colors in a centered coloring of $G$ is the treedepth of $G$, and it is known that classes of graphs with bounded treedepth are exactly those that exclude a fixed path as a subgraph, or equivalently, as a minor. In this sense, the structure of graphs excluding a fixed ladder as a minor resembles the structure of graphs without long paths. Another similarity is as follows: It is an easy observation that every connected graph with two vertex-disjoint paths of length $k$ has a path of length $k+1$. We show that every $3$-connected graph which contains as a minor a union of sufficiently many vertex-disjoint copies of a $2 \times k$ grid has a $2 \times (k+1)$ grid minor. Our structural results have applications to poset dimension. We show that posets whose cover graphs exclude a fixed ladder as a minor have bounded dimension. This is a new step towards the goal of understanding which graphs are unavoidable as minors in cover graphs of posets with large dimension.

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arXiv2021

Obstructions for bounded shrub-depth and rank-depth

Shrub-depth and rank-depth are dense analogues of the tree-depth of a graph. It is well known that a graph has large tree-depth if and only if it has a long path as a subgraph. We prove an analogous statement for shrub-depth and rank-depth, which was conjectured by Hliněný, Kwon, Obdržálek, and Ordyniak [Tree-depth and vertex-minors, European J.~Combin. 2016]. Namely, we prove that a graph has large rank-depth if and only if it has a vertex-minor isomorphic to a long path. This implies that for every integer $t$, the class of graphs with no vertex-minor isomorphic to the path on $t$ vertices has bounded shrub-depth.

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arXiv2021

Quickly excluding a non-planar graph

A cornerstone theorem in the Graph Minors series of Robertson and Seymour is the result that every graph $G$ with no minor isomorphic to a fixed graph $H$ has a certain structure. The structure can then be exploited to deduce far-reaching consequences. The exact statement requires some explanation, but roughly it says that there exist integers $k,n$ depending on $H$ only such that $0<k<n$ and for every $n\times n$ grid minor $J$ of $G$ the graph $G$ has a a $k$-near embedding in a surface $Σ$ that does not embed $H$ in such a way that a substantial part of $J$ is embedded in $Σ$. Here a $k$-near embedding means that after deleting at most $k$ vertices the graph can be drawn in $Σ$ without crossings, except for local areas of non-planarity, where crossings are permitted, but at most $k$ of these areas are attached to the rest of the graph by four or more vertices and inside those the graph is constrained in a different way, again depending on the parameter $k$. The original and only proof so far is quite long and uses many results developed in the Graph Minors series. We give a proof that uses only our earlier paper [A new proof of the flat wall theorem, {\it J.~Combin.\ Theory Ser.\ B \bf 129} (2018), 158--203] and results from graduate textbooks. Our proof is constructive and yields a polynomial time algorithm to construct such a structure. We also give explicit constants for the structure theorem, whereas the original proof only guarantees the existence of such constants.

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arXiv2019

A unified Erdős-Pósa theorem for constrained cycles

A doubly group-labeled graph is an oriented graph with its edges labeled by elements of the direct sum of two groups $Γ_1,Γ_2$. A cycle in a doubly group-labeled graph is $(Γ_1,Γ_2)$-non-zero if it is non-zero in both coordinates. Our main result is a generalization of the Flat Wall Theorem of Robertson and Seymour to doubly group-labeled graphs. As an application, we determine all canonical obstructions to the Erdős-Pósa property for $(Γ_1,Γ_2)$-non-zero cycles in doubly group-labeled graphs. The obstructions imply that the half-integral Erdős-Pósa property always holds for $(Γ_1,Γ_2)$-non-zero cycles. Moreover, our approach gives a unified framework for proving packing results for constrained cycles in graphs. For example, as immediate corollaries we recover the Erdős-Pósa property for cycles and $S$-cycles and the half-integral Erdős-Pósa property for odd cycles and odd $S$-cycles. Furthermore, we recover Reed's Escher-wall Theorem. We also prove many new packing results as immediate corollaries. For example, we show that the half-integral Erdős-Pósa property holds for cycles not homologous to zero, odd cycles not homologous to zero, and $S$-cycles not homologous to zero. Moreover, the (full) Erdős-Pósa property holds for $S_1$-$S_2$-cycles and cycles not homologous to zero on an orientable surface. Finally, we also describe the canonical obstructions to the Erdős-Pósa property for cycles not homologous to zero and for odd $S$-cycles.

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arXiv2018

Chi-boundedness of graph classes excluding wheel vertex-minors

A class of graphs is $χ$-bounded if there exists a function $f:\mathbb N\rightarrow \mathbb N$ such that for every graph $G$ in the class and an induced subgraph $H$ of $G$, if $H$ has no clique of size $q+1$, then the chromatic number of $H$ is less than or equal to $f(q)$. We denote by $W_n$ the wheel graph on $n+1$ vertices. We show that the class of graphs having no vertex-minor isomorphic to $W_n$ is $χ$-bounded. This generalizes several previous results; $χ$-boundedness for circle graphs, for graphs having no $W_5$ vertex-minors, and for graphs having no fan vertex-minors.

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arXiv2017

Nonrepetitive colourings of graphs excluding a fixed immersion or topological minor

We prove that graphs excluding a fixed immersion have bounded nonrepetitive chromatic number. More generally, we prove that if $H$ is a fixed planar graph that has a planar embedding with all the vertices with degree at least 4 on a single face, then graphs excluding $H$ as a topological minor have bounded nonrepetitive chromatic number. This is the largest class of graphs known to have bounded nonrepetitive chromatic number.

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arXiv2017

Forcing clique immersions through chromatic number

Building on recent work of Dvořák and Yepremyan, we show that every simple graph of minimum degree $7t+7$ contains $K_t$ as an immersion and that every graph with chromatic number at least $3.54t + 4$ contains $K_t$ as an immersion. We also show that every graph on $n$ vertices with no stable set of size three contains $K_{2\lfloor n/5 \rfloor}$ as an immersion.

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arXiv2016

Half-integral linkages in highly connected directed graphs

We study the half-integral $k$-Directed Disjoint Paths Problem ($\tfrac12$kDDPP) in highly strongly connected digraphs. The integral kDDPP is NP-complete even when restricted to instances where $k=2$, and the input graph is $L$-strongly connected, for any $L\geq 1$. We show that when the integrality condition is relaxed to allow each vertex to be used in two paths, the problem becomes efficiently solvable in highly connected digraphs (even with $k$ as part of the input). Specifically, we show that there is an absolute constant $c$ such that for each $k\geq 2$ there exists $L(k)$ such that $\tfrac12$kDDPP is solvable in time $O(|V(G)|^c)$ for a $L(k)$-strongly connected directed graph $G$. As the function $L(k)$ grows rather quickly, we also show that $\tfrac12$kDDPP is solvable in time $O(|V(G)|^{f(k)})$ in $(36k^3+2k)$-strongly connected directed graphs. We also show that for each $ε<1$ deciding half-integral feasibility of kDDPP instances is NP-complete when $k$ is given as part of the input, even when restricted to graphs with strong connectivity $εk$.

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arXiv2016

A New Proof of the Flat Wall Theorem

We give an elementary and self-contained proof, and a numerical improvement, of a weaker form of the excluded clique minor theorem of Robertson and Seymour, the following. Let t,r>0 be integers, and let R=49152t^{24}(40t^2+r). An r-wall is obtained from a (2r x r)-grid by deleting every odd vertical edge in every odd row and every even vertical edge in every even row, then deleting the two resulting vertices of degree one, and finally subdividing edges arbitrarily. The vertices of degree two that existed before the subdivision are called the pegs of the r-wall. Let G be a graph with no K_t minor, and let W be an R-wall in G. We prove that there exist a subset A of V(G) of size at most 12288t^{24} and an r-subwall W' of W such that V(W') is disjoint from A and W' is a flat wall in G-A in the following sense. There exists a separation (X,Y) of G-A such that X\cap Y is a subset of the vertex set of the cycle C' that bounds the outer face of W', V(W') is a subset of Y, every peg of W' belongs to X and the graph G[Y] can almost be drawn in the unit disk with the vertices X\cap Y drawn on the boundary of the disk in the order determined by C'. Here almost means that the assertion holds after repeatedly removing parts of the graph separated from X\cap Y by a cutset Z of size at most three, and adding all edges with both ends in Z. Our proof gives rise to an algorithm that runs in polynomial time even when r and t are part of the input instance. The proof is self-contained in the sense that it uses only results whose proofs can be found in textbooks.

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