arXiv subjects
Publications and source records attributed to Zdeněk Dvořák.
The first open case of Hadwiger's conjecture states that every $K_7$-minor-free graph is 6-colorable. We prove that this is the case for $K_7^=$-minor-free graphs, where $K_7^=$ denotes the graph obtained from $K_7$ by deleting two independent edges. The proof is based on an independently interesting density result: Every 5-connected $K_7^=$-minor-free graph with $n\ge 6$ vertices has at most $4n-8$ edges.
We show that if an n-vertex 5-connected graph has at least 4n-10 edges, then for any choice of five of its vertices, we can contract disjoint connected subgraphs containing these vertices to obtain $K_5$ as a minor. The bound on the number of edges is the best possible.
We study $c$-crossing-critical graphs, which are the minimal graphs that require at least $c$ edge-crossings when drawn in the plane. For $c=1$ there are only two such graphs without degree-2 vertices, $K_5$ and $K_{3,3}$, but for any fixed $c>1$ there exist infinitely many $c$-crossing-critical graphs. It has been previously shown that $c$-crossing-critical graphs have bounded path-width and contain only a bounded number of internally disjoint paths between any two vertices. We expand on these results, providing a more detailed description of the structure of crossing-critical graphs. On the way towards this description, we prove a new structural characterisation of plane graphs of bounded path-width. Then we show that every $c$-crossing-critical graph can be obtained from a $c$-crossing-critical graph of bounded size by replicating bounded-size parts that already appear in narrow "bands" or "fans" in the graph. This also gives an algorithm to generate all the $c$-crossing-critical graphs of at most given order $n$ in polynomial time per each generated graph.
Borodin and Kostochka in 1977 conjectured that if a graph $G$ has maximum degree $Δ(G)\ge 9$ and its clique number satisfies $ω(G)\le Δ(G)-1$, then its chromatic number satisfies $χ(G) \le Δ(G)-1$. We prove this statement with respect to a stronger graph coloring parameter, the correspondence chromatic number $χ_{DP}$, provided the maximum degree is sufficiently large. More precisely, we prove that for every integer $Δ\ge 3\cdot 10^9$, a graph $G$ of maximum degree at most $Δ$ satisfies $χ_{DP}(G) \le \max(ω(G),Δ-1)$. This strengthens earlier results of Reed (1999) for usual chromatic number and of Choi, Kierstead and Rabern (2023) for list chromatic number.
Lovász et al. proved that every $6$-edge-connected graph has a nowhere-zero $3$-flow. In fact, they proved a more technical statement which says that there exists a nowhere zero $3$-flow that extends the flow prescribed on the incident edges of a single vertex $z$ with bounded degree. We extend this theorem of Lovász et al. to allow $z$ to have arbitrary degree, but with the additional assumption that there is another vertex $x$ with large degree and no small cut separating $x$ and $z$. Using this theorem, we prove two results regarding the generation of minimal graphs with the property that prescribing the edges incident to a vertex with specific flow does not extend to a nowhere-zero $3$-flow. We use this to further strengthen the theorem of Lovász et al., as well as make progress on a conjecture of Li et al.
We give an exact characterization of monotone graph classes C with bounded average degree that satisfy the following property: The domination number of every graph from C is bounded by a linear function of its 2-independence number.
Since its beginnings, every Cycles and Colourings workshop holds one or two open problem sessions; this document contains the problems (together with notes regarding the current state of the art and related bibliography) presented by participants of the 33rd edition of the workshop which took place in Nový Smokovec, Slovakia during August 31st - September 5th, 2025 (see the workshop webpage https://candc.upjs.sk).
In 1980, Gupta, Kahn and Robertson proved that every graph $G$ with minimum degree at least $k\geq 2$ contains a cycle $C$ containing at least $k+1$ vertices each having at least $k$ neighbors in $C$ (so $C$ has at least $\frac{(k+1)(k-2)}{2}$ chords). In this work, we go further by showing that some of its edges can be contracted to obtain a graph with high minimum degree (we call such a minor of $C$ a \emph{cyclic minor}). We then investigate further cycles having cliques as cyclic minors, and show that minimum degree at least $O(k^2)$ guarantees a cyclic $K_k$-minor.
The assignments of a set of $m$ items into $n$ clusters of prescribed sizes $k_1,\dots,k_n$ can be encoded as the vertices of the partition polytope $\mathrm{PP}(k_1,\dots,k_n)$. We prove that, if $K = \max\{k_1,\dots,k_n\}$, then the combinatorial diameter of $\mathrm{PP}(k_1,\dots,k_n)$ is at most $\lceil 3K/2\rceil$. This improves the previously known upper bound of $2K$. A cycle (or path) odd-cover of a graph $G$ is a set of cycles (or paths) with symmetric difference $G$. We prove that every Eulerian graph $G$ with maximum degree $Δ$ admits a cycle odd-cover and a path odd-cover, each of size at most $\lceil 3Δ/4\rceil$. This improves the previously known upper bound of $Δ$. The two proofs share many similarities and are both based on the proof of Akiyama, Exoo, and Harary that every graph with maximum degree 4 has linear arboricity at most 3.
We conjecture that every graph of minimum degree five with no separating triangles and drawn in the plane with one crossing is 4-colorable. In this paper, we use computer enumeration to show that this conjecture holds for all graphs with at most 28 vertices, explore the consequences of this conjecture and provide some insights on how it could be proved.
DeVos and Seymour (2003) proved that for every set $C$ of 3-colorings of a set $X$ of vertices, there exists a plane graph $G$ with vertices of $X$ incident with the outer face such that a 3-coloring of $X$ extends to a 3-coloring of $G$ if and only if it belongs to $C$. We prove a generalization of this claim for $k$-colorings of $X$-rooted-$K_{k+1}$-minor-free $K_{k+2}$-minor-free graphs.
We prove that up to two exceptions, every connected subcubic triangle-free graph has fractional chromatic number at most 11/4. This is tight unless further exceptional graphs are excluded, and improves the known bound on the fractional chromatic number of subcubic triangle-free planar graphs.
Through computer-assisted enumeration, we list minimal obstructions for 5-choosability of graphs on the torus with the following additional property: There exists a cyclic system of non-contractible triangles around the torus where the consecutive triangles are at distance at most four. This condition is satisfied by all previously known obstructions, and we verify that there are no additional obstructions with this property. This supports the conjecture that a toroidal graph is 5-choosable if and only if it is 5-colorable.
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.
We consider the 4-precoloring extension problem in \emph{planar near-Eulerian-triangulations}, i.e., plane graphs where all faces except possibly for the outer one have length three, all vertices not incident with the outer face have even degree, and exactly the vertices incident with the outer face are precolored. We give a necessary topological condition for the precoloring to extend, and give a complete characterization when the outer face has length at most five and when all vertices of the outer face have odd degree and are colored using only three colors.
We investigate a fundamental vertex-deletion problem called (Induced) Subgraph Hitting: given a graph $G$ and a set $\mathcal{F}$ of forbidden graphs, the goal is to compute a minimum-sized set $S$ of vertices of $G$ such that $G-S$ does not contain any graph in $\mathcal{F}$ as an (induced) subgraph. This is a generic problem that encompasses many well-known problems that were extensively studied on their own, particularly (but not only) from the perspectives of both approximation and parameterization. In this paper, we study the approximability of the problem on a large variety of graph classes. Our first result is a linear-time $(1+\varepsilon)$-approximation reduction from (Induced) Subgraph Hitting on any graph class $\mathcal{G}$ of bounded expansion to the same problem on bounded degree graphs within $\mathcal{G}$. This directly yields linear-size $(1+\varepsilon)$-approximation lossy kernels for the problems on any bounded-expansion graph classes. Our second result is a linear-time approximation scheme for (Induced) Subgraph Hitting on any graph class $\mathcal{G}$ of polynomial expansion, based on the local-search framework of Har-Peled and Quanrud [SICOMP 2017]. This approximation scheme can be applied to a more general family of problems that aim to hit all subgraphs satisfying a certain property $π$ that is efficiently testable and has bounded diameter. Both of our results have applications to Subgraph Hitting (not induced) on wide classes of geometric intersection graphs, resulting in linear-size lossy kernels and (near-)linear time approximation schemes for the problem.
We show that $\limsup |E(G)|/|V(G)| = 2.5$ over all $4$-critical planar graphs $G$, answering a question of Grünbaum from 1988.
We investigate the product structure of hereditary graph classes admitting strongly sublinear separators. We characterise such classes as subgraphs of the strong product of a star and a complete graph of strongly sublinear size. In a more precise result, we show that if any hereditary graph class $\mathcal{G}$ admits $O(n^{1-ε})$ separators, then for any fixed $δ\in(0,ε)$ every $n$-vertex graph in $\mathcal{G}$ is a subgraph of the strong product of a graph $H$ with bounded tree-depth and a complete graph of size $O(n^{1-ε+δ})$. This result holds with $δ=0$ if we allow $H$ to have tree-depth $O(\log\log n)$. Moreover, using extensions of classical isoperimetric inequalties for grids graphs, we show the dependence on $δ$ in our results and the above $\text{td}(H)\in O(\log\log n)$ bound are both best possible. We prove that $n$-vertex graphs of bounded treewidth are subgraphs of the product of a graph with tree-depth $t$ and a complete graph of size $O(n^{1/t})$, which is best possible. Finally, we investigate the conjecture that for any hereditary graph class $\mathcal{G}$ that admits $O(n^{1-ε})$ separators, every $n$-vertex graph in $\mathcal{G}$ is a subgraph of the strong product of a graph $H$ with bounded tree-width and a complete graph of size $O(n^{1-ε})$. We prove this for various classes $\mathcal{G}$ of interest.