Motivated by Hadwiger's conjecture, we prove that every $n$-vertex graph $G$ with no independent set of size three contains an $\lceil n/2\rceil$-vertex simple minor $H$ with $$0.98688 \cdot \binom{|V(H)|}{2} - o(n^2)$$ edges.

Product structure theory aims to understand complex graphs by embedding them into products of simpler graphs. In this direction, Campbell, Distel, Gollin, Harvey, Hendrey, Hickingbotham, Mohar and Wood (2022) put forth the conjecture that all graphs of degree-d polynomial growth (i.e., where balls of radius r have 𝒪(r^d) vertices) can be embedded into the strong product of d trees, each with linear growth, and a constant-size clique. In this paper, we disprove this conjecture for d = 4. The counterexamples are finite subgraphs of a Cayley graph of the discrete 3-dimensional Heisenberg group ℍ(ℤ). These graphs were first proposed by Huang and McCarty as potential counterexamples to the conjecture. A key technical tool of our proof is the ”quantitative central collapse” theorem due to Cheeger, Kleiner and Naor (2011), guaranteeing that every Lipschitz map from the continuous Heisenberg group ℍ to the function space L_1 collapses along a central line.

Motivated by an old question of Gallai (1966) on the intersection of longest paths in a graph and the famous conjectures of Lov & aring;sz (1969) and Thomassen (1978) on the maximum length of paths and cycles in vertextransitive graphs, we present improved bounds for the parameters lpt(G) and lct(G), defined as the minimum size of a set of vertices in a graph G hitting all longest paths (cycles, respectively). First, we show that every connected graph G on n vertices satisfies lpt(G) <= root 8n, and lct(G) <= root 8n if G is additionally 2-connected. This improves a sequence of earlier bounds for these problems, with the previous state of the art being O(n(2/3)). Second, we show that every connected graph G satisfies lpt(G) <= O(l(5/9)), where l denotes the maximum length of a path in G. As an application of this latter bound, we present further progress towards Lov & aring;sz' and Thomassen's conjectures: We show that every connected vertex-transitive graph of order n contains a cycle (and path) of length Omega(n(9/14)). This improves the previous best bound of the form Omega(n(13/21)). Interestingly, our proofs employ several concepts and results from structural graph theory, such as a result of Robertson and Seymour [J. Combin. Theory Ser. B 49 (1990), pp. 40-77] on transactions in societies and Tutte's 2-separator theorem.

A "dominating K_t-model" in a graph G is a sequence (T_1,…,T_t) of pairwise vertex-disjoint connected subgraphs of G, such that whenever 1≤ i<j≤ t every vertex in T_j has a neighbour in T_i. Replacing "every vertex in T_j" by "some vertex in T_j" retrieves the standard definition of K_t-model, which is equivalent to a K_t-minor in G. We prove that every graph with no dominating K_5-model is 4-colourable. This generalises and is significantly stronger than the 4-colour theorem for planar graphs or for graphs with no K_5-minor. It also makes progress towards Hajós' conjecture on K_5-subdivisions in 5-chromatic graphs.

Given a graph $G$ and an integer $d\ge 0$, its $d$-defective chromatic number $\chi^d(G)$ is the smallest size of a partition of the vertices into parts inducing subgraphs with maximum degree at most $d$. Guo, Kang and Zwaneveld recently studied the relationship between the $d$-defective chromatic number of the $(d+1)$-fold (clique) blowup $G\boxtimes K_{d+1}$ of a graph $G$ and its ordinary chromatic number, and conjectured that $\chi(G)=\chi^d(G\boxtimes K_{d+1})$ for every graph $G$ and $d\ge 0$. In this note we disprove this conjecture by constructing graphs $G$ of arbitrarily large chromatic number such that $\chi(G)\ge \frac{30}{29}\chi^d(G\boxtimes K_{d+1})$ for infinitely many $d$. On the positive side, we show that the conjecture holds with a constant factor correction, namely $\chi^d(G\boxtimes K_{d+1})\le \chi(G)\le 2\chi^d(G\boxtimes K_{d+1})$ for every graph $G$ and $d\ge 0$.

Illingworth and Wood recently proposed the Dominating Hadwiger's Conjecture, a strengthening of Hadwiger's Conjecture which asserts that every graph with no dominating K_t-model is (t-1)-colorable. We prove two relaxations of this conjecture. First, we show that every graph with average degree Ct (log t)^2 contains a dominating K_t-model for some absolute constant C. This bound improves on the 2^t-2 due to Illingworth and Wood and is within an O(log t) factor from optimal. Second, we prove that the vertices of every graph with no dominating K_t-model can be partitioned into t-1 parts such that the subgraph induced by each part has bounded maximum degree.

An edge-coloring of a graph G assigns a color to each edge of G. An edge-coloring is a parity edge-coloring if for each path P in G, it uses some color on an odd number of edges in P. It is a strong parity edge-coloring if for every open walk W in G, it uses some color an odd number of times along W. The minimum numbers of colors in parity and strong parity edge-colorings of G are denoted p(G) and p(G), respectively. We characterize strong parity edge-colorings and use this to prove lower bounds on p(G) and answer several questions of Bunde, Milans, West, and Wu. The applications are as follows. (1) We prove the conjecture that p(K-s,K-t) = s o t, where s o t is the Hopf-Stiefel function. (2) We show that p(G) for a connected n-vertex graph G equals the known lower bound [log(2) n] if and only if G is a subgraph of the hypercube Q[log(2)n]. (3) We asymptotically compute p(G) when G is the & ell;th distance-power of a path, proving p(P-n(& ell;)) similar to & ell; [log(2)n]. (4) We disprove the conjecture that p(G) = p(G) when G is bipartite by constructing bipartite graphs G such that p(G)/p(G) is arbitrarily large; in particular, with p(G) >= 1-o(1)/3 k ln k and p(G) <= 2k + k(1/3).

We show that intersection graphs of compact convex sets in Rn of bounded aspect ratio have asymptotic dimension at most 2n+1. More generally, we show this is the case for intersection graphs of systems of subsets of any metric space of Assouad–Nagata dimension n that satisfy the following condition: For each r,s>0 and every point p, the number of pairwise-disjoint elements of diameter at least s in the system that are at distance at most r from p is bounded by a function of r/s.

Motivated by Hadwiger's conjecture, we study the problem of finding the densest possible $t$-vertex minor in graphs of average degree at least $t-1$. We show that if $G$ has average degree at least $t-1$, it contains a minor on $t$ vertices with at least $(\sqrt{2}-1-o(1))\binom{t}{2}$ edges. We show that this cannot be improved beyond $\left(\frac{3}{4}+o(1)\right)\binom{t}{2}$. Finally, for $t\leq 6$ we exactly determine the number of edges we are guaranteed to find in the densest $t$-vertex minor in graphs of average degree at least $t-1$.

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 connected graph G is 3-flow-critical if G does not have a nowhere-zero 3-flow, but every proper contraction of G does. We prove that every n-vertex 3-flow-critical graph other than K2 and K4 has at least 53n edges. This bound is tight up to lower-order terms, answering a question of Li et al. (2022). It also generalizes the result of Koester (1991) on the maximum average degree of 4-critical planar graphs.

Motivated by Hadwiger's conjecture, Seymour asked which graphs $H$ have the property that every non-null graph $G$ with no $H$ minor has a vertex of degree at most $|V(H)|-2$. We show that for every monotone graph family $\mathcal{F}$ with strongly sublinear separators, all sufficiently large bipartite graphs $H \in \mathcal{F}$ with bounded maximum degree have this property. None of the conditions that $H$ belongs to $\mathcal{F}$, that $H$ is bipartite and that $H$ has bounded maximum degree can be omitted.

We show that every $n$-vertex planar graph is 3-colourable with monochromatic components of size $O(n^{4/9})$. The best previous bound was $O(n^{1/2})$ due to Linial, Matoušek, Sheffet and Tardos [Combin. Probab. Comput., 2008].

We initiate the study of nowhere-zero flow reconfiguration. The natural question is whether any two nowhere-zero k-flows of a given graph G are connected by a sequence of nowhere-zero k-flows of G, such that any two consecutive flows in the sequence differ only on a cycle of G. We study this problem in the setting of integer flows and group flows, and prove a number of positive and negative results. * The natural reconfiguration variant of Tutte's 5-flow conjecture, stating that any two nowhere-zero 5-flows in any 2-edge-connected graph are connected, is false in the group and integer cases. * All nowhere-zero ℤ_2^8-flows of every 2-edge-connected graph are connected and for every sufficiently large abelian group A, all nowhere-zero A-flows of every 2-edge-connected graph are connected. * The group structure affects the answer, contrary to the existence problem for nowhere-zero flows. * We highlight a duality with recoloring in planar graphs and deduce that any two nowhere-zero 7-flows in a planar graph are connected, among other results. * For every 2-edge-connected graph G, there is an integer k such that all nowhere-zero k-flows of G are connected.

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].

Let K_7^∨ denote the graph obtained from the complete graph on seven vertices by deleting two edges with a common end. Motivated by Hadwiger's conjecture, we prove that every graph with no K_7^∨-minor is 6-colorable.

Consider a graph $G$ drawn on a fixed surface, and assign to each vertex a list of colors of size at least two if $G$ is triangle-free and at least three otherwise. We prove that we can give each vertex a color from its list so that each monochromatic connected subgraph has bounded weak diameter (i.e., diameter measured in the metric of the whole graph $G$, not just the subgraph). In case that $G$ has bounded maximum degree, this implies that each connected monochromatic subgraph has bounded size. This solves a problem of Esperet and Joret for planar triangle-free graphs, and extends known results in the general case to the list setting, answering a question of Wood.

In a recent breakthrough Campos, Griffiths, Morris and Sahasrabudhe obtained the first exponential improvement of the upper bound on the diagonal Ramsey numbers since 1935. We shorten their proof, replacing the underlying book algorithm with a simple inductive statement. This modification allows us - to give a very short proof of an improved upper bound on the off-diagonal Ramsey numbers, which extends to the multicolor setting, and - to clarify the dependence of the bounds on underlying parameters and optimize these parameters, obtaining, in particular, an upper bound R(k,k) ≤ (3.8)^k+o(k) on the diagonal Ramsey numbers.

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 product of two graphs each with bounded treewidth. For graphs excluding a fixed minor, we strengthen the result by showing that every such graph is contained in the strong product of two digraphs with bounded indegree and with bounded treewidth. This result has the advantage that the product now has bounded degeneracy. In the setting of 3-term products, we show that every $K_t$-minor-free graph is contained in $H_1\boxtimes H_2 \boxtimes K_{c(t)}$ where $\text{tw}(H_i)\leq t-2$. This treewidth bound is close to tight: in any such result with $\text{tw}(H_i)$ bounded, both $H_1$ and $H_2$ can be forced to contain any graph of treewidth $t-5$, implying $\text{tw}(H_1)\geq t-5$ and $\text{tw}(H_2)\geq t-5$. Analogous lower and upper bounds are shown for any excluded minor, where the minimum possible bound on $\text{tw}(H_i)$ is tied to the treedepth of the excluded minor. Subgraphs of the product of two graphs with bounded treewidth have two tree-decompositions where any bag from the first decomposition intersects any bag from the second decomposition in a bounded number, $k$, of vertices, so called $k$-orthogonal tree-decompositions. We show that graphs excluding a fixed odd-minor have a tree-decomposition and a path-decomposition that are $O(1)$-orthogonal. This implies that such graphs have a tree-decomposition in which each bag has bounded pathwidth. This result is best possible in that `pathwidth' cannot be replaced by `bandwidth' or `treedepth'. Moreover, we characterize the minor-closed classes that have a tree-decomposition in which each bag has bounded bandwidth, or each bag has bounded treedepth.

The burning number b(G) of a graph G is the smallest number of turns required to burn all vertices of a graph if at every turn a new fire is started and existing fires spread to all adjacent vertices. The Burning Number Conjecture of Bonato et al. (2016) postulates that b(G)≤⌈n⌉ for all connected graphs G on n vertices. We prove that this conjecture holds asymptotically, that is b(G)≤(1+o(1))n.