Related papers: Minors and dimension
A fundamental result in structural graph theory states that every graph with large average degree contains a large complete graph as a minor. We prove this result with the extra property that the minor is small with respect to the order of…
We prove the Erd\H os--S\'os conjecture for trees with bounded maximum degree and large dense host graphs. As a corollary, we obtain an upper bound on the multicolour Ramsey number of large trees whose maximum degree is bounded by a…
We study the relations between several notions of dimension for an additive set, some of which are well-known and some of which are more recent, appearing for instance in work of Schoen and Shkredov. We obtain bounds for the ratios between…
We discuss a possible characterization, by means of forbidden configurations, of posets which are embeddable in a product of finitely many scattered chains.
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 Dujmovi\'c, Eppstein, Joret, Morin, and Wood (SIDMA,…
Let $G$ be a graph, and let $u$, $v$, and $w$ be vertices of $G$. If the distance between $u$ and $w$ does not equal the distance between $v$ and $w$, then $w$ is said to resolve $u$ and $v$. The metric dimension of $G$, denoted $\beta(G)$,…
Layered treewidth and row treewidth are recently introduced graph parameters that have been key ingredients in the solution of several well-known open problems. It follows from the definitions that the layered treewidth of a graph is at…
We study large minors in small-set expanders. More precisely, we consider graphs with $n$ vertices and the property that every set of size at most $\alpha n / t$ expands by a factor of $t$, for some (constant) $\alpha > 0$ and large $t =…
We prove that the tree-width of graphs in a hereditary class defined by a finite set $F$ of forbidden induced subgraphs is bounded if and only if $F$ includes a complete graph, a complete bipartite graph, a tripod (a forest in which every…
We use a classical result of McCord and reduction methods of finite spaces to prove a generalization of Thomason's theorem on homotopy colimits over posets. In particular this allows us to characterize the homotopy colimits of diagrams of…
A \emph{signed graph} $(G, \sigma)$ is a graph $G$ together with an assignment $\sigma:E(G) \rightarrow \{+,-\}$. The notion of homomorphisms of signed graphs is a relatively new development which allows to strengthen the connection between…
In this article we study a coarse version of the K-theoretic Farrell-Jones conjecture we call coarse or bounded isomorphism conjecture. With techniques that have already been used to prove the Farrell-Jones conjecture for hyperbolic groups…
We prove new bounds on the dimensions of distance sets and pinned distance sets of planar sets. Among other results, we show that if $A\subset\mathbb{R}^2$ is a Borel set of Hausdorff dimension $s>1$, then its distance set has Hausdorff…
A recent result of Chepoi, Estellon and Vaxes [DCG '07] states that any planar graph of diameter at most 2R can be covered by a constant number of balls of size R; put another way, there are a constant-sized subset of vertices within which…
This paper explores the structure of graphs defined by an excluded minor or an excluded odd minor through the lens of graph products and tree-decompositions. We prove that every graph excluding a fixed odd minor is contained in the strong…
The scramble number of a graph provides a lower bound for gonality and an upper bound for treewidth, making it a graph invariant of interest. In this paper we study graphs of scramble number at most two, and give a classification of all…
In recent years, there has been significant interest in characterizing the induced subgraph obstructions to bounded treewidth and pathwidth. While this has recently been resolved for pathwidth, the case of treewidth remains open, and prior…
We consider hereditary classes of graphs equipped with a total order. We provide multiple equivalent characterisations of those classes which have bounded twin-width. In particular, we prove a grid theorem for classes of ordered graphs…
The planar graph product structure theorem of Dujmovi\'{c}, Joret, Micek, Morin, Ueckerdt, and Wood [J. ACM 2020] states that every planar graph is a subgraph of the strong product of a graph with bounded treewidth and a path. This result…
It has been conjectured by Golumbic and Monma in 1984 that the intersection of tolerance and cocomparability graphs coincides with bounded tolerance graphs. The conjecture has been proved under some - rather strong - \emph{structural}…