A subgraph-universal graph/a topological minor-universal graph in a class of graphs $\mathcal{G}$ is a graph in $\mathcal{G}$ which contains every graph in $\mathcal{G}$ as a subgraph/topological minor. We prove that the class $\mathcal{P}$ of all countable planar graphs does not contain a topological minor-universal graph. This answers a question of Diestel and Kühn and strengthens a result of Pach stating that there is no subgraph-universal graph in $\mathcal{P}$. Furthermore, we characterise for which subdivided stars $T$ there is a topological minor-universal graph in the class of all countable $T$-free graphs.
Huynh et al. recently showed that a countable graph $G$ which contains every countable planar graph as a subgraph must contain arbitrarily large finite complete graphs as topological minors, and an infinite complete graph as a minor. We strengthen this result by showing that the same conclusion holds, if $G$ contains every countable planar graph as a topological minor. In particular, there is no countable planar graph containing every countable planar graph as a topological minor, answering a question by Diestel and Kühn. Moreover, we construct a locally finite planar graph which contains ever…
A graph $U$ is universal for a graph class $\mathcal{C}\ni U$, if every $G\in \mathcal{C}$ is a minor of $U$. We prove the existence or absence of universal graphs in several natural graph classes, including graphs component-wise embeddable into a surface, and graphs forbidding $K_5$, or $K_{3,3}$, or $K_\infty$ as a minor. We prove the existence of uncountably many minor-closed classes of countable graphs that (do and) do not have a universal element. Some of our results and questions may be of interest to the finite graph theorist. In particular, one of our side-results is that every $K_5$-m…
Mader proved that for every integer $t$ there is a smallest real number $c(t)$ such that any graph with average degree at least $c(t)$ must contain a $K_t$-minor. Fiorini, Joret, Theis and Wood conjectured that any graph with $n$ vertices and average degree at least $c(t)+ε$ must contain a $K_t$-minor consisting of at most $C(ε,t)\log n$ vertices. Shapira and Sudakov subsequently proved that such a graph contains a $K_t$-minor consisting of at most $C(ε,t)\log n \log\log n$ vertices. Here we build on their method using graph expansion to remove the $\log\log n$ factor and prove the conjecture.…
As a major step in their proof of Wagner's conjecture, Robertson and Seymour showed that every graph not containing a fixed graph $H$ as a minor has a tree-decomposition in which each torso is almost embeddable in a surface of bounded genus. Recently, Grohe and Marx proved a similar result for graphs not containing $H$ as a topological minor. They showed that every graph which does not contain $H$ as a topological minor has a tree-decomposition in which every torso is either almost embeddable in a surface of bounded genus, or has a bounded number of vertices of high degree. We give a short pro…
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.
We show that posets of bounded height whose cover graphs exclude a fixed graph as a topological minor have bounded dimension. This result was already proven by Walczak. However, our argument is entirely combinatorial and does not rely on structural decomposition theorems. Given a poset with large dimension but bounded height, we directly find a large clique subdivision in its cover graph. Therefore, our proof is accessible to readers not familiar with topological graph theory, and it allows us to provide explicit upper bounds on the dimension. With the introduced tools we show a second result …
In this paper, we consider a structural and geometric property of graphs, namely the presence of large expanders. The problem of finding such structures was first considered by Krivelevich [SIAM J. Disc. Math. 32 1 (2018)]. Here, we show that the problem of finding a large induced subgraph that is an expander can be reduced to the simpler problem of finding a topological minor that is an expander. Our proof is constructive, which is helpful in an algorithmic setting. We also show that every large subgraph of an expander graph contains a large subgraph which is itself an expander.
We give a complete combinatorial characterization of homogeneous quadratic relations of "universal character" valid for minors of quantum matrices (more precisely, for minors in the quantized coordinate ring $O_q(M_{m,n}(K))$ of $m\times n$ matrices over a field $K$, where $q\in K^\ast$). This is obtained as a consequence of a study of quantized minors of matrices generated by paths in certain planar graphs, called SE-graphs, generalizing the ones associated with Cauchon diagrams. Our efficient method of verifying universal quadratic identities for minors of quantum matrices is illustrated wit…
Given graphs $G$ and $H$, we propose a method to implicitly enumerate topological-minor-embeddings of $H$ in $G$ using decision diagrams. We show a useful application of our method to enumerating subgraphs characterized by forbidden topological minors, that is, planar, outerplanar, series-parallel, and cactus subgraphs. Computational experiments show that our method can find all planar subgraphs in a given graph at most five orders of magnitude faster than a naive backtracking-based method.
This sequel to our paper (Infinite gammoids, 2014) considers minors and duals of infinite gammoids. We prove that a class of gammoids definable by digraphs not containing a certain type of substructure, called an outgoing comb, is minor-closed. Also, we prove that finite-rank minors of gammoids are gammoids. Furthermore, the topological gammoids introduced by Carmesin (Topological infinite gammoids, and a new Menger-type theorem for infinite graphs, 2014) are proved to coincide, as matroids, with the finitary gammoids. A corollary is that topological gammoids are minor-closed. It is a well-kno…
This paper continues an earlier research of the authors on universal quadratic identities (QIs) on minors of quantum matrices. We demonstrate situations when the universal QIs are provided, in a sense, by the ones of four special types (Plucker, co-Plucker, Dodgson identities and quasi-commutation relations on flag and co-flag interval minors).
Institute for Basic Science（基础科学研究所）；Simon Fraser University（西蒙菲莎大学）；Charles University（查理大学）；Technical University of Denmark（丹麦技术大学）；Monash University（莫纳什大学）
A well-known theorem of Nash-Williams shows that the collection of locally finite trees under the topological minor relation results in a BQO. Set theoretically, two very natural questions arise: (1) What is the number $λ$ of topological types of locally finite trees? (2) What are the possible sizes of an equivalence class of locally finite trees? For (1), clearly, $ω\leq λ\leq \mathfrak{c}$ and Matthiesen refined it to $ω_1 \leq λ\leq \mathfrak{c}$. Thus, this question becomes non-trivial when the Continuum Hypothesis is not assumed. In this paper we address both questions by showing that - e…
Extremal properties of sparse graphs, randomly perturbed by the binomial random graph are considered. It is known that every $n$-vertex graph $G$ contains a complete minor of order $Ω(n/α(G))$. We prove that adding $ξn$ random edges, where $ξ> 0$ is arbitrarily small yet fixed, to an $n$-vertex graph $G$ satisfying $α(G) \leq ζ(ξ)n$ asymptotically almost surely results in a graph containing a complete minor of order $\tilde Ω\left( n/\sqrt{α(G)}\right)$; this result is tight up to the implicit logarithmic terms. For complete topological minors, we prove that there exists a constant $C>0$ such …
The Hadwiger number $h(G)$ is the order of the largest complete minor in $G$. Does sufficient Hadwiger number imply a minor with additional properties? In [2], Geelen et al showed $h(G)\geq (1+o(1))ct\sqrt{\ln t}$ implies $G$ has a bipartite subgraph with Hadwiger number at least $t$, for some explicit $c\sim 1.276\dotsc$. We improve this to $h(G) \geq (1+o(1))t\sqrt{\log_2 t}$, and provide a construction showing this is tight. We also derive improved bounds for the topological minor variant of this problem.
For a finite collection of graphs ${\cal F}$, the \textsc{${\cal F}$-TM-Deletion} problem has as input an $n$-vertex graph $G$ and an integer $k$ and asks whether there exists a set $S \subseteq V(G)$ with $|S| \leq k$ such that $G \setminus S$ does not contain any of the graphs in ${\cal F}$ as a topological minor. We prove that for every such ${\cal F}$, \textsc{${\cal F}$-TM-Deletion} is fixed parameter tractable on planar graphs. Our algorithm runs in a $2^{\mathcal{O}(k^2)}\cdot n^{2}$ time or, alternatively in $2^{\mathcal{O}(k)}\cdot n^{4}$ time. Our techniques can easily be extended to…
Warsaw University of Technology（华沙理工大学）；University of Warsaw（华沙大学）；FAMNIT and IAM, University of Primorska（普里莫斯卡大学 FAMNIT 与 IAM）；IT University of Copenhagen（哥本哈根信息技术大学）
It is proved that for every $\varepsilon>0$, there exists $K>0$ such that for every integer $t\ge2$, every graph with chromatic number at least $Kt$ contains a minor with $t$ vertices and edge density at least $1-\varepsilon$. Indeed, building on recent work of Delcourt and Postle on linear Hadwiger's conjecture, for $\varepsilon\in(0,\frac{1}{256})$ we can take $K=C\log\log(1/\varepsilon)$ where $C>0$ is a universal constant, which extends their recent $O(t\log\log t)$ bound on the chromatic number of graphs with no $K_t$ minor.
Let $V_{8}+e$ denote the unique graph obtained from the Wagner graph, also known as $V_{8}$, by adding an edge between two vertices of distance 3 on the Hamilton cycle, which is exactly a split of a minor of the Petersen graph. A complete characterization of all internally 4-connected graphs with no $V_{8}$ minor is given in [J. Maharry and N. Robertson, The structure of graphs not topologically containing the Wagner graph, J. Combin. Theory Ser. B 121 (2016) 398-420]. In this paper we characterize all internally 4-connected graphs with no $V_{8}+e$ minor.
We show that any self-complementary graph with $n$ vertices contains a $K_{\lfloor \frac{n+1}{2}\rfloor}$ minor. We derive topological properties of self-complementary graphs.
For random graphs, the containment problem considers the probability that a binomial random graph $G(n,p)$ contains a given graph as a substructure. When asking for the graph as a topological minor, i.e., for a copy of a subdivision of the given graph, it is well-known that the (sharp) threshold is at $p=1/n$. We consider a natural analogue of this question for higher-dimensional random complexes $X^k(n,p)$, first studied by Cohen, Costa, Farber and Kappeler for $k=2$. Improving previous results, we show that $p=Θ(1/\sqrt{n})$ is the (coarse) threshold for containing a subdivision of any fixed…
Let $\triangleleft$ be a relation between graphs. We say a graph $G$ is \emph{$\triangleleft$-ubiquitous} if whenever $Γ$ is a graph with $nG \triangleleft Γ$ for all $n \in \mathbb{N}$, then one also has $\aleph_0 G \triangleleft Γ$, where $αG$ is the disjoint union of $α$ many copies of $G$. The \emph{Ubiquity Conjecture} of Andreae, a well-known open problem in the theory of infinite graphs, asserts that every locally finite connected graph is ubiquitous with respect to the minor relation. In this paper, which is the first of a series of papers making progress towards the Ubiquity Conjectur…
We prove an asymptotically tight bound on the extremal density guaranteeing subdivisions of bounded-degree bipartite graphs with a mild separability condition. As corollaries, we answer several questions of Reed and Wood on embedding sparse minors. Among others, $\bullet$ $(1+o(1))t^2$ average degree is sufficient to force the $t\times t$ grid as a topological minor; $\bullet$ $(3/2+o(1))t$ average degree forces every $t$-vertex planar graph as a minor, and the constant $3/2$ is optimal, furthermore, surprisingly, the value is the same for $t$-vertex graphs embeddable on any fixed surface; $\b…
For a graph class $\mathcal{F}$, let $ex_{\mathcal{F}}(n)$ denote the maximum number of edges in a graph in $\mathcal{F}$ on $n$ vertices. We show that for every proper minor-closed graph class $\mathcal{F}$ the function $ex_{\mathcal{F}}(n) - Δn$ is eventually periodic, where $Δ= \lim_{n \to \infty} ex_{\mathcal{F}}(n)/n$ is the limiting density of $\mathcal{F}$. This confirms a special case of a conjecture by Geelen, Gerards and Whittle. In particular, the limiting density of every proper minor-closed graph class is rational, which answers a question of Eppstein. As a major step in the proof…
Archdeacon (1987) proved that graphs embeddable on a fixed surface can be $3$-coloured so that each colour class induces a subgraph of bounded maximum degree. Edwards, Kang, Kim, Oum and Seymour (2015) proved that graphs with no $K_{t+1}$-minor can be $t$-coloured so that each colour class induces a subgraph of bounded maximum degree. We prove a common generalisation of these theorems with a weaker assumption about excluded subgraphs. This result leads to new defective colouring results for several graph classes, including graphs with linear crossing number, graphs with given thickness (with r…
In this paper we show that every graph of pathwidth less than $k$ that has a path of order $n$ also has an induced path of order at least $\frac{1}{3} n^{1/k}$. This is an exponential improvement and a generalization of the polylogarithmic bounds obtained by Esperet, Lemoine and Maffray (2016) for interval graphs of bounded clique number. We complement this result with an upper-bound. This result is then used to prove the two following generalizations: - every graph of treewidth less than $k$ that has a path of order $n$ contains an induced path of order at least $\frac{1}{4} (\log n)^{1/k}$; …
We give a signed generalization of Laurent's theorem that characterizes feasible positive semidefinite matrix completion problems in terms of metric polytopes. Based on this result, we give a characterization of the maximum rank completions of the signed positive semidefinite matrix completion problem for odd-$K_4$ minor free signed graphs. The analysis can also be used to bound the minimum rank over the completions and to characterize uniquely solvable completion problems for odd-$K_4$ minor free signed graphs. As a corollary we derive a characterization of the universal rigidity of odd-$K_4$…
Let $G$ be a simple graph on the vertex set $\{1,\ldots,n\}$ with $m$ edges. An algebraic object attached to $G$ is the ideal $P_{G}$ generated by diagonal 2-minors of an $n \times n$ matrix of variables. In this paper we prove that if $G$ is bipartite, then every initial ideal of $P_{G}$ is generated by squarefree monomials of degree at most $\left \lfloor{\frac{m+n+1}{2}} \right \rfloor$. Furthermore, we completely characterize all connected graphs $G$ for which $P_{G}$ is the toric ideal associated to a finite simple graph. Finally we compute in certain cases the universal Gr{ö}bner basis o…
It has been known for 30 years that posets with bounded height and with cover graphs of bounded maximum degree have bounded dimension. Recently, Streib and Trotter proved that dimension is bounded for posets with bounded height and planar cover graphs, and Joret et al. proved that dimension is bounded for posets with bounded height and with cover graphs of bounded tree-width. In this paper, it is proved that posets of bounded height whose cover graphs exclude a fixed topological minor have bounded dimension. This generalizes all the aforementioned results and verifies a conjecture of Joret et …
We prove that posets of bounded height whose cover graphs belong to a fixed class with bounded expansion have bounded dimension. Bounded expansion, introduced by Nešetřil and Ossona de Mendez as a model for sparsity in graphs, is a property that is naturally satisfied by a wide range of graph classes, from graph structure theory (graphs excluding a minor or a topological minor) to graph drawing (e.g. graphs with bounded book thickness). Therefore, our theorem generalizes a number of results including the most recent one for posets of bounded height with cover graphs excluding a fixed graph as …
