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Register a new userUniversality in minor-closed graph classes
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Stanislaw Ulam asked whether there exists a universal countable planar graph (that is, a countable planar graph that contains every countable planar graph as a subgraph). Janos Pach (1981) answered this question in the negative. We strengthen this result by showing that every countable graph that contains all countable planar graphs must contain (i) an infinite complete graph as a minor, and (ii) a subdivision of the complete graph $K_t$ with multiplicity $t$, for every finite $t$. On the other hand, we construct a countable graph that contains all countable planar graphs and has several key properties such as linear colouring numbers, linear expansion, and every finite $n$-vertex subgraph has a balanced separator of size $O(sqrt{n})$. The graph is $mathcal{T}_6boxtimes P_{!infty}$, where $mathcal{T}_k$ is the universal treewidth-$k$ countable graph (which we define explicitly), $P_{!infty}$ is the 1-way infinite path, and $boxtimes$ denotes the strong product. More generally, for every positive integer $t$ we construct a countable graph that contains every countable $K_t$-minor-free graph and has the above key properties. Our final contribution is a construction of a countable graph that contains every countable $K_t$-minor-free graph as an induced subgraph, has linear colouring numbers and linear expansion, and contains no subdivision of the countably infinite complete graph (implying (ii) above is best possible).
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Binary functions are a generalisation of the cocircuit spaces of binary matroids to arbitrary functions. Every rank function is assigned a binary function, and the deletion and contraction operations of binary functions generalise matroid deletion and contraction. We give the excluded minor characterisations for the classes of binary functions with well defined minors, and those with an associated rank function. Within these classes, we also characterise the classes of binary functions corresponding to polymatroids, matroids and binary matroids by their excluded minors. This gives a new proof of Tuttes excluded minor characterisation of binary matroids in the more generalised space of binary functions.
Let $mathcal G$ be an addable, minor-closed class of graphs. We prove that the zero-one law holds in monadic second-order logic (MSO) for the random graph drawn uniformly at random from all {em connected} graphs in $mathcal G$ on $n$ vertices, and the convergence law in MSO holds if we draw uniformly at random from all graphs in $mathcal G$ on $n$ vertices. We also prove analogues of these results for the class of graphs embeddable on a fixed surface, provided we restrict attention to first order logic (FO). Moreover, the limiting probability that a given FO sentence is satisfied is independent of the surface $S$. We also prove that the closure of the set of limiting probabilities is always the finite union of at least two disjoint intervals, and that it is the same for FO and MSO. For the classes of forests and planar graphs we are able to determine the closure of the set of limiting probabilities precisely. For planar graphs it consists of exactly 108 intervals, each of length $approx 5cdot 10^{-6}$. Finally, we analyse examples of non-addable classes where the behaviour is quite different. For instance, the zero-one law does not hold for the random caterpillar on $n$ vertices, even in FO.
Let $t$ be a positive real number. A graph is called $t$-tough, if the removal of any cutset $S$ leaves at most $|S|/t$ components. The toughness of a graph is the largest $t$ for which the graph is $t$-tough. A graph is minimally $t$-tough, if the toughness of the graph is $t$ and the deletion of any edge from the graph decreases the toughness. In this paper we investigate the minimum degree and the recognizability of minimally $t$-tough graphs in the class of chordal graphs, split graphs, claw-free graphs and $2K_2$-free graphs.
What is the maximum number of copies of a fixed forest $T$ in an $n$-vertex graph in a graph class $mathcal{G}$ as $nto infty$? We answer this question for a variety of sparse graph classes $mathcal{G}$. In particular, we show that the answer is $Theta(n^{alpha_d(T)})$ where $alpha_d(T)$ is the size of the largest stable set in the subforest of $T$ induced by the vertices of degree at most $d$, for some integer $d$ that depends on $mathcal{G}$. For example, when $mathcal{G}$ is the class of $k$-degenerate graphs then $d=k$; when $mathcal{G}$ is the class of graphs containing no $K_{s,t}$-minor ($tgeq s$) then $d=s-1$; and when $mathcal{G}$ is the class of $k$-planar graphs then $d=2$. All these results are in fact consequences of a single lemma in terms of a finite set of excluded subgraphs.
The asymptotic dimension is an invariant of metric spaces introduced by Gromov in the context of geometric group theory. In this paper, we study the asymptotic dimension of metric spaces generated by graphs and their shortest path metric and show their applications to some continuous spaces. The asymptotic dimension of such graph metrics can be seen as a large scale generalisation of weak diameter network decomposition which has been extensively studied in computer science. We prove that every proper minor-closed family of graphs has asymptotic dimension at most 2, which gives optimal answers to a question of Fujiwara and Papasoglu and (in a strong form) to a problem raised by Ostrovskii and Rosenthal on minor excluded groups. For some special minor-closed families, such as the class of graphs embeddable in a surface of bounded Euler genus, we prove a stronger result and apply this to show that complete Riemannian surfaces have Assouad-Nagata dimension at most 2. Furthermore, our techniques allow us to prove optimal results for the asymptotic dimension of graphs of bounded layered treewidth and graphs of polynomial growth, which are graph classes that are defined by purely combinatorial notions and properly contain graph classes with some natural topological and geometric flavours.
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