arXiv subjects

Sergey Norin

Publications and source records attributed to Sergey Norin.

At least 19 recordsLinked to original sources

arXiv2026

Two Relaxations of the Dominating Hadwiger's Conjecture

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.

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arXiv2026

Nerve-type and invariance theorems for asymptotic dimension

Asymptotic dimension of metric spaces is a large-scale analog of covering dimension of topological spaces. An intersection graph of a family of sets is the graph whose vertices are the members of the family and whose edges correspond to pairs of members with non-empty intersection. Our first main result connects the asymptotic dimension of the intersection graph of a family ${\mathcal F}$ and the Assouad-Nagata dimension of the ambient metric space containing members of ${\mathcal F}$ under some mild and necessary assumptions. We prove that if ${\mathcal F}$ is a family of subsets of a metric space of Assouad-Nagata dimension $n$ such that every ball of radius $r$ intersects at most $f(r/s)$ pairwise disjoint members of ${\mathcal F}$ of diameter at least $s$ for some function $f$, then the asymptotic dimension of the intersection graph of ${\mathcal F}$ is at most $n+1$. This result is optimal both quantitatively and qualitatively in several senses. As a corollary of this result, the asymptotic dimension of the intersection graph of any family of compact convex sets of bounded aspect ratio in ${\mathbb R}^n$, such as a family of balls in ${\mathbb R}^n$, is at most $n+1$. Our second main result states that the asymptotic dimension of the intersection graph of a family ${\mathcal F}$ of connected closed sets of a connected topological space with connected boundary equals the asymptotic dimension of the intersection graph of the family of the boundary of the sets in ${\mathcal F}$, under a mild condition. In particular, the asymptotic dimension of the intersection graphs of families of spheres in ${\mathbb R}^n$ equals $n$ or $n+1$ when $n \geq 2$.

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arXiv2026

Disproof of the tree product conjecture via the Heisenberg group

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 $\mathcal{O}(r^d)$ 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 $\mathbb{H}(\mathbb{Z})$. 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 $\mathbb{H}$ to the function space $L_1$ collapses along a central line.

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arXiv2026

The Dominating 4-Colour Theorem

A "dominating $K_t$-model" in a graph $G$ is a sequence $(T_1,\dots,T_t)$ of pairwise vertex-disjoint connected subgraphs of $G$, such that whenever $1\leq i<j\leq t$ 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 Haj\'{o}s' conjecture on $K_5$-subdivisions in $5$-chromatic graphs.

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arXiv2026

Trade-off between spread and width for tree decompositions

The spread of a vertex $v$ in a tree decomposition is the number of bags that contain $v$. We study the trade-off between spread and width in tree decompositions, answering every open question from Wood [arXiv:2509.01140]. First, Wood asked for the infimum of $c > 0$ such that there exists $c'$ such that each graph $G$ has a tree decomposition of width $c \cdot tw(G)$ in which each vertex $v$ has spread at most $c'(d(v)+1)$. We show that the answer is $3$. Second, we prove a conjecture of Wood, stating that every tree-decomposition of the $(n \times n)$-grid with width $n$ has a vertex with spread $\Omega(n)$. Finally, we answer the last question of Wood by showing that near-optimal average spread can be achieved simultaneously with width $O(tw(G))$.

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arXiv2025

Nowhere-zero flow reconfiguration

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 $\mathbb{Z}_2^8$-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.

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arXiv2025

Every graph with no $K_7^{\vee}$-minor is $6$-colorable

Let $K_7^{\vee}$ 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 $K_7^{\vee}$-minor is $6$-colorable.

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arXiv2025

3-Colouring Planar Graphs

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, Matou\v{s}ek, Sheffet and Tardos [Combin. Probab. Comput., 2008].

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arXiv2025

Small hitting sets for longest paths and cycles

Motivated by an old question of Gallai (1966) on the intersection of longest paths in a graph and the well-known conjectures of Lov\'{a}sz (1969) and Thomassen (1978) on the maximum length of paths and cycles in vertex-transitive graphs, we present improved bounds for the parameters $\mathrm{lpt}(G)$ and $\mathrm{lct}(G)$, 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 $\mathrm{lpt}(G)\le \sqrt{8n}$, and $\mathrm{lct}(G)\le \sqrt{8n}$ 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 $\mathrm{lpt}(G)\le O(\ell^{5/9})$, where $\ell$ denotes the maximum length of a path in $G$. As an immediate application of this latter bound, we present further progress towards Lov\'{a}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 make use of several concepts and results from structural graph theory, such as a result of Robertson and Seymour (1990) on transactions in societies and Tutte's $2$-separator theorem.

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arXiv2025

Defective coloring of blowups

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 $d\ge 0$.

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arXiv2024

Strong parity edge-colorings of graphs

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 $\hat{p}(G)$, respectively. We characterize strong parity edge-colorings and use this characterization to prove lower bounds on $\hat{p}(G)$ and answer several questions of Bunde, Milans, West, and Wu. The applications are as follows. (1) We prove the conjecture that $\hat{p}(K_{s,t})=s \circ t$, where $s \circ t$ is the Hopf-Stiefel function. (2) We show that $\hat{p}(G)$ for a connected $n$-vertex graph $G$ equals the known lower bound $\lceil \log_2 n \rceil$ if and only if $G$ is a subgraph of the hypercube $Q_{\lceil \log_2 n \rceil }$. (3) We asymptotically compute $\hat{p}(G)$ when $G$ is the $\ell$th distance-power of a path, proving $\hat{p}(P_n^\ell)\sim\ell \lceil {\log_2 n} \rceil$. (4) We disprove the conjecture that $\hat{p}(G)=p(G)$ when $G$ is bipartite by constructing bipartite graphs $G$ such that $\hat{p}(G)/p(G)$ is arbitrarily large; in particular, with $\hat{p}(G)\ge\frac{1-o(1)}3 k\ln k$ and $p(G)\le2k+k^{1/3}$.

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arXiv2024

Product Structure and Tree-Decompositions

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.

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arXiv2024

Optimizing the CGMS upper bound on Ramsey numbers

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 $$R(k,k) \leq (3.8)^{k+o(k)}$$ on the diagonal Ramsey numbers.

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arXiv2024

Sparsity of 3-flow critical graphs

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 K_2 and K_4 has at least 5n/3 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.

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arXiv2023

Twin-width of sparse random graphs

We show that the twin-width of every $n$-vertex $d$-regular graph is at most $n^{\frac{d-2}{2d-2}+o(1)}$ and that almost all $d$-regular graphs attain this bound. More generally, we obtain bounds on the twin-width of sparse Erd\H{o}s-Renyi and regular random graphs, complementing the bounds in the denser regime due to Ahn, Chakraborti, Hendrey, Kim and Oum.

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arXiv2023

On an induced version of Menger's theorem

We prove Menger-type results in which the obtained paths are pairwise non-adjacent, both for graphs of bounded maximum degree and, more generally, for graphs excluding a topological minor. We further show better bounds in the subcubic case, and in particular obtain a tight result for two paths using a computer-assisted proof.

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arXiv2023

Finding dense minors using average degree

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 $t-1$.

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arXiv2023

On the density of matroids omitting a complete-graphic minor

We show that, if $M$ is a simple rank-$n$ matroid with no $\ell$-point line minor and no minor isomorphic to the cycle matroid of a $t$-vertex complete graph, then the ratio $|M| / n$ is bounded above by a singly exponential function of $\ell$ and $t$. We also bound this ratio in the special case where $M$ is a frame matroid, obtaining an answer that is within a factor of two of best-possible.

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