A graph $G$ is said to be ubiquitous, if every graph $Gamma$ that contains arbitrarily many disjoint $G$-minors automatically contains infinitely many disjoint $G$-minors. The well-known Ubiquity conjecture of Andreae says that every locally finite g
raph is ubiquitous. In this paper we show that locally finite graphs admitting a certain type of tree-decomposition, which we call an extensive tree-decomposition, are ubiquitous. In particular this includes all locally finite graphs of finite tree-width, and also all locally finite graphs with finitely many ends, all of which have finite degree. It remains an open question whether every locally finite graph admits an extensive tree-decomposition.
We investigate games played between Maker and Breaker on an infinite complete graph whose vertices are coloured with colours from a given set, each colour appearing infinitely often. The players alternately claim edges, Makers aim being to claim all
edges of a sufficiently colourful infinite complete subgraph and Breakers aim being to prevent this. We show that if there are only finitely many colours then Maker can obtain a complete subgraph in which all colours appear infinitely often, but that Breaker can prevent this if there are infinitely many colours. Even when there are infinitely many colours, we show that Maker can obtain a complete subgraph in which infinitely many of the colours each appear infinitely often.
A {em connectivity function} on a set $E$ is a function $lambda:2^Erightarrow mathbb R$ such that $lambda(emptyset)=0$, that $lambda(X)=lambda(E-X)$ for all $Xsubseteq E$, and that $lambda(Xcap Y)+lambda(Xcup Y)leq lambda(X)+lambda(Y)$ for all $X,Y s
ubseteq E$. Graphs, matroids and, more generally, polymatroids have associated connectivity functions. In this paper we give a method for identifying when a connectivity function comes from a graph. This method uses no more than a polynomial number of evaluations of the connectivity function. In contrast, we show that the problem of identifying when a connectivity function comes from a matroid cannot be solved in polynomial time. We also show that the problem of identifying when a connectivity function is not that of a matroid cannot be solved in polynomial time.
A hypergroup is stringent if $a boxplus b$ is a singleton whenever $a eq -b$. A hyperfield is stringent if the underlying additive hypergroup is. Every doubly distributive skew hyperfield is stringent, but not vice versa. We present a classification
of stringent hypergroups, from which a classification of doubly distributive skew hyperfields follows. It follows from our classification that every such hyperfield is a quotient of a skew field.
It is known that the cop number $c(G)$ of a connected graph $G$ can be bounded as a function of the genus of the graph $g(G)$. The best known bound, that $c(G) leq leftlfloor frac{3 g(G)}{2}rightrfloor + 3$, was given by Schr{o}der, who conjectured t
hat in fact $c(G) leq g(G) + 3$. We give the first improvement to Schr{o}ders bound, showing that $c(G) leq frac{4g(G)}{3} + frac{10}{3}$.
We investigate valuated matroids with an additional algebraic structure on their residue matroids. We encode the structure in terms of representability over stringent hyperfields. A hyperfield $H$ is {em stringent} if $aboxplus b$ is a singleton un
less $a=-b$, for all $a,bin H$. By a construction of Marc Krasner, each valued field gives rise to a stringent hyperfield. We show that if $H$ is a stringent skew hyperfield, then the vectors of any weak matroid over $H$ are orthogonal to its covectors, and we deduce that weak matroids over $H$ are strong matroids over $H$. Also, we present vector axioms for matroids over stringent skew hyperfields which generalize the vector axioms for oriented matroids and valuated matroids.
Abstract separation systems are a new unifying framework in which separations of graph, matroids and other combinatorial structures can be expressed and studied. We characterize the abstract separation systems that have representations as separation systems of graphs, sets, or set bipartitions.
A graph $G$ is said to be $preceq$-ubiquitous, where $preceq$ is the minor relation between graphs, if whenever $Gamma$ is a graph with $nG preceq Gamma$ for all $n in mathbb{N}$, then one also has $aleph_0 G preceq Gamma$, where $alpha G$ is the dis
joint union of $alpha$ many copies of $G$. A well-known conjecture of Andreae is that every locally finite connected graph is $preceq$-ubiquitous. In this paper we give a sufficient condition on the structure of the ends of a graph~$G$ which implies that $G$ is $preceq$-ubiquitous. In particular this implies that the full grid is $preceq$-ubiquitous.
The class of quasi-graphic matroids recently introduced by Geelen, Gerards, and Whittle generalises each of the classes of frame matroids and lifted-graphic matroids introduced earlier by Zaslavsky. For each biased graph $(G, mathcal B)$ Zaslavsky de
fined a unique lift matroid $L(G, mathcal B)$ and a unique frame matroid $F(G, mathcal B)$, each on ground set $E(G)$. We show that in general there may be many quasi-graphic matroids on $E(G)$ and describe them all. We provide cryptomorphic descriptions in terms of subgraphs corresponding to circuits, cocircuits, independent sets, and bases. Equipped with these descriptions, we prove some results about quasi-graphic matroids. In particular, we provide alternate proofs that do not require 3-connectivity of two results of Geelen, Gerards, and Whittle for 3-connected matroids from their introductory paper: namely, that every quasi-graphic matroid linearly representable over a field is either lifted-graphic or frame, and that if a matroid $M$ has a framework with a loop that is not a loop of $M$ then $M$ is either lifted-graphic or frame. We also provide sufficient conditions for a quasi-graphic matroid to have a unique framework. Zaslavsky has asked for those matroids whose independent sets are contained in the collection of independent sets of $F(G, mathcal B)$ while containing those of $L(G, mathcal B)$, for some biased graph $(G, mathcal B)$. Adding a natural (and necessary) non-degeneracy condition defines a class of matroids, which we call biased graphic. We show that the class of biased graphic matroids almost coincides with the class of quasi-graphic matroids: every quasi-graphic matroid is biased graphic, and if $M$ is a biased graphic matroid that is not quasi-graphic then $M$ is a 2-sum of a frame matroid with one or more lifted-graphic matroids.
Let $triangleleft$ be a relation between graphs. We say a graph $G$ is emph{$triangleleft$-ubiquitous} if whenever $Gamma$ is a graph with $nG triangleleft Gamma$ for all $n in mathbb{N}$, then one also has $aleph_0 G triangleleft Gamma$, where $alph
a G$ is the disjoint union of $alpha$ 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 Conjecture, we show that all trees are ubiquitous with respect to the topological minor relation, irrespective of their cardinality. This answers a question of Andreae from 1979.
