We prove two results relating the basis number of a graph G to path decompositions of G. Our first result shows that the basis number of a graph is at most four times its pathwidth. Our second result shows that, if a graph G has a path decomposition with adhesions of size at most k in which the graph induced by each bag has basis number at most b, then G has basis number at most b+O(klog^2 k). The first result, combined with recent work of Geniet and Giocanti shows that the basis number of a graph is bounded by a polynomial function of its treewidth. The second result (also combined with the work of Geniet and Giocanti) shows that every K_t-minor-free graph has a basis number bounded by a polynomial function of t.

It is a notorious open question whether integer programs (IPs) with an integer coefficient matrix M whose subdeterminants are all bounded by a constant triangle in absolute value can be solved in polynomial time. We answer this question in the affirmative if we further require that, by removing a constant number of rows and columns from M, one obtains a submatrix A that is the transpose of a network matrix. Our approach focuses on the case in which A arises from M after removing k rows only and k is a constant. We achieve our result in two main steps, the first related to the theory of IPs and the second related to graph minor theory. First, we derive a strong proximity result for the case in which A is a general totally unimodular matrix: given an optimal solution of the linear programming relaxation, an optimal solution to the IP can be obtained by finding a constant number of augmentations by circuits of [A I]. Second, for the case in which A is the transpose of a network matrix, we reformulate the problem as a maximum constrained integer potential problem on a graph G. We observe that, if G is two-connected, then it has no rooted K2, t-minor for t = ohm(k triangle). We leverage this to obtain a tree-decomposition of G into highly structured graphs for which we can solve the problem locally. This allows us to solve the global problem via dynamic programming.

A family F of graphs is asymptotically χ-bounded with bounding function f if almost every graph G in the family satisfies χ(G) ≤ f(ω(G)). A graph is H-free if it does not contain H as an induced subgraph. We ask which hereditary families are asymptotically χ-bounded, and discuss some related questions. We show that for every tree T, almost all T-free graphs G satisfy χ(G)=ω(G). We show that for every cycle C_k except C_6, almost every C_k-free graph G satisfies χ(G) = ω(G). We show that the C_6-free graphs are asymptotically χ-bounded with bounding function f(w)=(1+o(1))w^2/log w.

We prove that for every tree T which is not an edge, for almost every graph G which does not contain T as an induced subgraph, V(G) has a partition into α(T)-1 parts certifying this fact. Each part induces a graph which is P_4-free and has further properties which depend on T. As a consequence we obtain good bounds (often tight up to a constant factor) on the number of T-free graphs and show in a follow-up paper that almost every T-free graph G has chromatic number equal to the size of its largest clique.

We explore the concept of separating systems of vertex sets of graphs. A separating system of a set X is a collection of subsets of X such that for any pair of distinct elements in X, there exists a set in the separating system that contains exactly one of the two elements. A separating system of the vertex set of a graph G is called a vertex-separating path (tree) system of G if the elements of the separating system are paths (trees) in the graph G. In this paper, we focus on the size of the smallest vertex-separating path (tree) system for different types of graphs, including trees, grids, and maximal outerplanar graphs.

The dominating number γ(G) of a graph G is the minimum size of a vertex set whose closed neighborhood covers all the vertices of the graph. The packing number ρ(G) of G is the maximum size of a vertex set whose closed neighborhoods are pairwise disjoint. In this paper we study graph classes G such that γ(G)/ρ(G) is bounded by a constant c_ G for each G∈ G. We propose an inductive proof technique to prove that if G is the class of 2-degenerate graphs, then there is such a constant bound c_ G. We note that this is the first monotone, dense graph class that is shown to have constant ratio. We also show that the classes of AT-free and unit-disk graphs have bounded ratio. In addition, our technique gives improved bounds on c_ G for planar graphs, graphs of bounded treewidth or bounded twin-width. Finally, we provide some new examples of graph classes where the ratio is unbounded.

A graph G is H-free if it does not contain an induced subgraph isomorphic to H. The study of the typical structure of H-free graphs was initiated by Erdős, Kleitman and Rothschild, who have shown that almost all C_3-free graphs are bipartite. Since then the typical structure of H-free graphs has been determined for several families of graphs H, including complete graphs, trees and cycles. Recently, Reed and Scott proposed a conjectural description of the typical structure of H-free graphs for all graphs H, which extends all previously known results in the area. We construct an infinite family of graphs for which the Reed-Scott conjecture fails, and use the methods we developed in the prequel paper to describe the typical structure of H-free graphs for graphs H in this family. Using similar techniques, we construct an infinite family of graphs H for which the maximum size of a homogenous set in a typical H-free graph is sublinear in the number of vertices, answering a question of Loebl et al. and Kang et al.

A rooted graph is a graph together with a designated vertex subset, called the roots. In this paper, we consider rooted graphs embedded in a fixed surface. A collection of faces of the embedding is a face cover if every root is incident to some face in the collection. We prove that every 3-connected, rooted graph that has no rooted K_2,t minor and is embedded in a surface of Euler genus g, has a face cover whose size is upper-bounded by some function of g and t, provided that the face-width of the embedding is large enough in terms of g. In the planar case, we prove an unconditional O(t^4) upper bound, improving a result of Böhme and Mohar . The higher genus case was claimed without a proof by Böhme, Kawarabayashi, Maharry and Mohar .

A family of graphs F is hereditary if F is closed under isomorphism and taking induced subgraphs. The speed of F is the sequence {|F-n|}(n is an element of N), where F-n denotes the set of graphs in F with the vertex set [n]. Alon, Balogh, Bollob & aacute;s and Morris [The structure of almost all graphs in a hereditary property, JCTB 2011] gave a rough description of typical graphs in a hereditary family and used it to show for every proper hereditary family F there exist epsilon > 0 and an integer l >= 1 such that |F-n|=2((1-1/l)n2/2+o(n2-epsilon)) The main result of this paper gives a more precise description of typical structure for a restricted class of hereditary families. As a consequence we characterize hereditary families with the speed just above the threshold 2((1-1/l)n2/2), generalizing a result of Balogh and Butterfield [Excluding induced subgraphs: Critical graphs, RSA 2011].

We prove that for every positive integer d and forest F, the class of intersection graphs of axis-aligned boxes in ℝ^d with no induced F subgraph is (polynomially) χ-bounded.

In the total matching problem, one is given a graph $G$ with weights on the vertices and edges. The goal is to find a maximum weight set of vertices and edges that is the non-incident union of a stable set and a matching. We consider the natural formulation of the problem as an integer program (IP), with variables corresponding to vertices and edges. Let $M = M(G)$ denote the constraint matrix of this IP. We define $\Delta(G)$ as the maximum absolute value of the determinant of a square submatrix of $M$. We show that the total matching problem can be solved in strongly polynomial time provided $\Delta(G) \leq \Delta$ for some constant $\Delta \in \mathbb{Z}_{\ge 1}$. We also show that the problem of computing $\Delta(G)$ admits an FPT algorithm. We also establish further results on $\Delta(G)$ when $G$ is a forest.

We introduce and study conflict-free colourings of t-subsets in hypergraphs. In such colourings, one assigns colours to all subsets of vertices of cardinality t such that in any hyperedge of cardinality at least t there is a uniquely coloured t-subset. The case t=1 , i.e., vertex conflict-free colouring, is a well-studied notion that originates in the context of frequency allocation to antennas in cellular networks. It has caught the attention of researchers both from the combinatorial and algorithmic points of view. A special focus was given to hypergraphs arising in geometry. Already the case t=2 (i.e., colouring pairs) seems to present a new challenge. Many of the tools used for conflict-free colouring of geometric hypergraphs rely on hereditary properties of the underlying hypergraphs. When dealing with subsets of vertices, the properties do not pass to subfamilies of subsets. Therefore, we develop new tools, which might be of independent interest. For t=2 we show that for each of those “well-behaved” hypergraphs H, the hypergraph H' , whose hyperedges are obtained by taking all the unions of two hyperedges from H, admits a 2-subset conflict-free colouring with roughly the same number of colours as H. For example, we show that the ( [ n; 2 ]) pairs of points in any set P of n points in the plane can be coloured with O(log n) colours such that for any two discs d_1,d_2 in the plane with |(d_1∪ d_2)∩ P|≥ 2 there is a uniquely (in d_1∪ d_2 ) coloured pair.

We study a natural generalization of the classical ε -net problem (Haussler and Welzl in Discrete Comput. Geom. 2 (2), 127–151 (1987)), which we call the ε – t - net problem : Given a hypergraph on n vertices and parameters t and ε≥ t/n , find a minimum-sized family S of t -element subsets of vertices such that each hyperedge of size at least ε n contains a set in S . When t=1 , this corresponds to the ε -net problem. We prove that any sufficiently large hypergraph with VC-dimension d admits an ε – t -net of size O((d(1+log t)/ε)log (1/ε )) . For some families of geometrically-defined hypergraphs (such as the dual hypergraph of regions with linear union complexity), we prove the existence of O(1/ε) -sized ε – t -nets. We also present an explicit construction of ε – t -nets (including ε -nets) for hypergraphs with bounded VC-dimension. In comparison to previous constructions for the special case of ε -nets (i.e., for t=1 ), it does not rely on advanced derandomization techniques. To this end we introduce a variant of the notion of VC-dimension which is of independent interest. Finally, we use our techniques to generalize the notion of ε -approximation and to prove the existence of small-sized ε – t -approximations for sufficiently large hypergraphs with a bounded VC-dimension.

For $n\geq s> r\geq 1$ and $k\geq 2$, write $n \rightarrow (s)_{k}^r$ if every hyperedge coloring with $k$ colors of the complete $r$-uniform hypergraph on $n$ vertices has a monochromatic subset of size $s$. Improving upon previous results by M. Axenovich, A. Gyárfás, H. Liu, and D. Mubayi [ Discrete Math., 322 (2014), pp. 69--77] and P. Erdös, A. Hajnal, A. Máté, and R. Rado, [ Combinatorial set theory: Partition Relations for Cardinals, Elsevier, Amsterdam, 1984] we show that $if r \geq 3 and n \nrightarrow (s)_k^r, then 2^n \nrightarrow (s+1)_{k+3}^{r+1}.$ This improves some of the known lower bounds on multicolor hypergraph Ramsey numbers. Given a hypergraph $H=(V,E)$, we consider the Ramsey-like problem of coloring all $r$-subsets of $V$ such that no hyperedge of size $\geq r+1$ is monochromatic. We provide upper and lower bounds on the number of colors necessary in terms of the chromatic number $\chi(H)$. In particular we show that this number is $O(\log^{(r-1)} (r \chi(H)) + r)$, where $\log^{y}$ is the $\log$ function applied $y$ times.

Weak and strong coloring numbers are generalizations of the degeneracy of a graph, where for each natural number $k$, we seek a vertex ordering such every vertex can (weakly respectively strongly) reach in $k$ steps only few vertices with lower index in the ordering. Both notions capture the sparsity of a graph or a graph class, and have interesting applications in the structural and algorithmic graph theory. Recently, the first author together with McCarty and Norin observed a natural volume-based upper bound for the strong coloring numbers of intersection graphs of well-behaved objects in $\mathbb{R}^d$, such as homothets of a centrally symmetric compact convex object, or comparable axis-aligned boxes. In this paper, we prove upper and lower bounds for the $k$-th weak coloring numbers of these classes of intersection graphs. As a consequence, we describe a natural graph class whose strong coloring numbers are polynomial in $k$, but the weak coloring numbers are exponential. We also observe a surprising difference in terms of the dependence of the weak coloring numbers on the dimension between touching graphs of balls (single-exponential) and hypercubes (double-exponential).

We study a natural generalization of the classical $$\varepsilon $$ -net problem (Haussler and Welzl in Discrete Comput. Geom. 2(2), 127–151 (1987)), which we call the $$\varepsilon $$ –t-net problem: Given a hypergraph on n vertices and parameters t and $$\varepsilon \ge t/n$$ , find a minimum-sized family S of t-element subsets of vertices such that each hyperedge of size at least $$\varepsilon n$$ contains a set in S. When $$t=1$$ , this corresponds to the $$\varepsilon $$ -net problem. We prove that any sufficiently large hypergraph with VC-dimension d admits an $$\varepsilon $$ –t-net of size $$O(({d(1+\log t)}/{\varepsilon })\log (1/\varepsilon ))$$ . For some families of geometrically-defined hypergraphs (such as the dual hypergraph of regions with linear union complexity), we prove the existence of $$O({1}/{\varepsilon })$$ -sized $$\varepsilon $$ –t-nets. We also present an explicit construction of $$\varepsilon $$ –t-nets (including $$\varepsilon $$ -nets) for hypergraphs with bounded VC-dimension. In comparison to previous constructions for the special case of $$\varepsilon $$ -nets (i.e., for $$t=1$$ ), it does not rely on advanced derandomization techniques. To this end we introduce a variant of the notion of VC-dimension which is of independent interest. Finally, we use our techniques to generalize the notion of $$\varepsilon $$ -approximation and to prove the existence of small-sized $$\varepsilon $$ –t-approximations for sufficiently large hypergraphs with a bounded VC-dimension.

We give a strongly polynomial-time algorithm for integer linear programs defined by integer coefficient matrices whose subdeterminants are bounded by a constant and that contain at most two nonzero entries in each row. The core of our approach is the first polynomial-time algorithm for the weighted stable set problem on graphs that do not contain more than $k$ vertex-disjoint odd cycles, where $k$ is any constant. Previously, polynomial-time algorithms were only known for $k=0$ (bipartite graphs) and for $k=1$ . We observe that integer linear programs defined by coefficient matrices with bounded subdeterminants and two nonzeros per column can be also solved in strongly polynomial-time, using a reduction to b-matching.

We construct an infinite family of counterexamples to Thomassen's conjecture that the vertices of every 3-connected, cubic graph on at least 8 vertices can be colored blue and red such that the blue subgraph has maximum degree at most 1 and the red subgraph minimum degree at least 1 and contains no path on 4 vertices.

For n≥ s> r≥ 1 and k≥ 2, write n → (s)_k^r if every hyperedge colouring with k colours of the complete r-uniform hypergraph on n vertices has a monochromatic subset of size s. Improving upon previous results by AGLM14 and EHMR84 we show that if r ≥ 3 and n ↛ (s)_k^r then 2^n ↛ (s+1)_k+3^r+1. This yields an improvement for some of the known lower bounds on multicolour hypergraph Ramsey numbers. Given a hypergraph H=(V,E), we consider the Ramsey-like problem of colouring all r-subsets of V such that no hyperedge of size ≥ r+1 is monochromatic. We provide upper and lower bounds on the number of colours necessary in terms of the chromatic number χ(H). In particular we show that this number is O(log^(r-1) (r χ(H)) + r).

A string graph is the intersection graph of a family of continuous arcs in the plane. The intersection graph of a family of plane convex sets is a string graph, but not all string graphs can be obtained in this way. We prove the following structure theorem conjectured by Janson and Uzzell: The vertex set of almost all string graphs on n vertices can be partitioned into five cliques such that some pair of them is not connected by any edge (n ->infinity). We also show that every graph with the above property is an intersection graph of plane convex sets. As a corollary, we obtain that almost all string graphs on n vertices are intersection graphs of plane convex sets.