We give a near-linear time 4-coloring algorithm for planar graphs, improving on the previous quadratic time algorithm by Robertson et al. from 1996. Such an algorithm cannot be achieved by the known proofs of the Four Color Theorem (4CT). Technically speaking, we show the following significant generalization of the 4CT: every planar triangulation contains linearly many pairwise non-touching reducible configurations or pairwise non-crossing obstructing cycles of length at most 5 (which all allow for making effective 4-coloring reductions). The known proofs of the 4CT only show the existence of a single reducible configuration or obstructing cycle in the above statement. The existence is proved using the discharging method based on combinatorial curvature. It identifies reducible configurations in parts where the local neighborhood has positive combinatorial curvature. Our result significantly strengthens the known proofs of 4CT, showing that we can also find reductions in large “flat" parts where the curvature is zero, and moreover, we can make reductions almost anywhere in a given planar graph. An interesting aspect of this is that such large flat parts are also found in large triangulations of any fixed surface. From a computational perspective, the old proofs allowed us to apply induction on a problem that is smaller by some additive constant. The inductive step took linear time, resulting in a quadratic total time. With our linear number of reducible configurations or obstructing cycles, we can reduce the problem size by a constant factor. Our inductive step takes O(nlog n) time, yielding a 4-coloring in O(nlog n) total time. In order to efficiently handle a linear number of reducible configurations, we need them to have certain robustness that could also be useful in other applications. All our reducible configurations are what is known as D-reducible.

In every 3-connected graph, some longest cycle has a chord. (c) 2026 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).

The vertex set of any planar graph of minimum degree at least 3 can be colored in two colors so that every vertex has a neighbor of each color. If the graph is a planar triangulation, the coloring can be chosen such that every vertex has a neighbor of its own color and at least two neighbors of the opposite color.

We consider the following variant of the edge-augmentation problem: Given a kedge-connected graph with no loops or multiple edges, find a smallest edge set in the complement whose addition to G results in a (k+1)-edge-connected graph. We establish the following dichotomy for this problem: If the complement of G contains a matching covering all vertices of G-degree k (and possibly more), then the complement also contains a matching whose addition to G results in a (k+1)edge-connected graph. A smallest matching which augments the minimum degree can be found, in polynomial time, by Edmonds' matching algorithm, but it need not augment the edge-connectivity. Indeed, it is NP-hard to find a smallest edge-connectivity augmenting edge set, by a result of Tibor Jorda'\n. On the other hand, if the complement of G contains no matching covering all vertices of G-degree k, then the complement has a minimum degree augmenting path system consisting of paths of length 1 or 2. Again we can find such a path system with as few edges as possible by Edmonds' matching algorithm. We can, in polynomial time, modify it to an edge-connectivity augmenting path system of paths of length 1 or 2 with the same number of edges, and this time it yields a smallest edge-connectivity augmenting set of edges. Combining these results, we conclude that a smallest edge-connectivity augmenting edge set in the complement of a k-regular, k-edge-connected simple graph has size n - m(G), where n is the number of vertices of G, and m(G) is the size of a maximum matching in the complement of G. Another corollary is that the complement of every simple noncomplete graph G with n vertices has a set of at most 2n/3 edges whose addition to G results in a graph of larger edge-connectivity, with equality holding if and only the complement of G is a disjoint union of 3-cycles.

We initiate the study of chromatic numbers for contact graphs of configurations of integer-sized cuboids in three dimensions, all of which are mutually congruent. Disallowing rotations, we show a global upper bound of 8 for the chromatic numbers, which implies that there is a global upper bound of 48 when the cuboids may be rotated freely. Specializing further to cuboids that are required to have a side length of one we obtain more precise upper bounds. Such upper bounds are compared to examples of configurations having relatively large chromatic numbers, leading to a complete determination of some of these chromatic numbers, but in general, the gaps between our upper and lower bounds are rather wide. In particular, we know of no such configuration of any size leading to a chromatic number above 6.

For every natural number k, there exists a planar graph which is Z2k-colorable, but not Γ-colorable for any other Abelian group Γ of order 2k. Its dual graph is Z2k-connected, but not Γ-connected for any other Abelian group Γ of order 2k.

We describe an infinite family of strongly 2-connected oriented graphs (that is, directed graphs with no multiple arcs) containing no arc whose reversal decreases the number of directed cycles.

The number of proper vertex-3-colorings of every triangle-free planar graph with n vertices and with no separating cycle of length 4 or 5 is at least 2n/17700000. On the other hand, for infinitely many n, there exists a triangle-free planar graph with separating cycles of length 4 and 5 whose number of proper vertex-3-colorings is <215n/log2⁡(n).

A graph has arboricity α if its edges can be partitioned into α forests. The dynamic arboricity decomposition problem is to update a partitioning of the graph’s edges into forests, as a graph undergoes insertions and deletions of edges. We present an algorithm for maintaining partitioning into α + 1 forests, provided the arboricity of the dynamic graph never exceeds α . Our algorithm has an update time of ˜ O ( n 3 / 4 ) when α is at most polylogarithmic in n. Similarly, the dynamic bounded out-orientation problem is to orient the edges of the graph such that the out-degree of each vertex is at all times bounded. For this problem, we give an algorithm that orients the edges such that the out-degree is at all times bounded by α + 1, with an update time of ˜ O ( n 5 / 7 ), when α is at most polylogarithmic in n . Here, the choice of α + 1 should be viewed in the light of the well-known lower bound by Brodal and Fagerberg which establishes that, for general graphs, maintaining only α out-edges would require linear update time. However, the lower bound by Brodal and Fagerberg is non-planar. In this paper, we give a lower bound showing that even for planar graphs, linear update time is needed in order to maintain an explicit three-out-orientation. For planar graphs, we show that the dynamic four forest decomposition and four-out-orientations, can be updated in ˜ O ( n 1 / 2 ) time.

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). J\'anos 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 $\mathcal{T}_6\boxtimes P_{\!\infty}$, 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).

Every planar simple graph with n vertices has at least 2n/9 Z5-colorings.

We combine matrix theory and graph theory methods to give a complete characterization of the surjective linear transformations of tropical matrices that preserve the cyclicity index. We show that there are non-surjective linear transformations that preserve the cyclicity index and we leave it open to characterize those.

In this paper we investigate density conditions for finding a complete $r$-uniform hypergraph $K_{r+1}^{(r)}$ on $r+1$ vertices in an $(r+1)$-partite $r$-uniform hypergraph $G$. First we prove an optimal condition in terms of the densities of the $(r+1)$ induced $r$-partite subgraphs of $G$. Second, we prove a version of this result where we assume that $r$-tuples of vertices in $G$ have their neighbours evenly distributed in $G$. Third, we also prove a counting result for the minimum number of copies of $K_{r+1}^{(r)}$ when $G$ satisfies our density bound, and present some open problems. A striking difference between the graph, $r=2$, and the hypergraph, $ r \geq 3 $, cases is that in the first case both the existence threshold and the counting function are non-linear in the involved densities, whereas for hypergraphs they are given by a linear function. Also, the smallest density of the $r$-partite parts needed to ensure the existence of a complete $r$-graph with $(r+1)$ vertices is equal to the golden ratio $\tau=0.618\ldots$ for $r=2$, while it is $\frac{r}{r+1}$for $r\geq3$.

We prove that there exists an infinite family of 4-regular 4-connected Hamiltonian graphs with a bounded number of Hamiltonian cycles. We do not know if there exists such a family of 5-regular 5-connected Hamiltonian graphs.

The number of cycles in a graph containing any fixed edge and also containing all vertices of odd degree is odd if and only if all vertices have even degree. If all vertices have even degree this is a theorem of Shunichi Toida. If all vertices have odd degree it is Andrew Thomason's extension of Smith's theorem.

Every 9-regular graph (possibly with multiple edges) with odd edge-connectivity >5 can be edge-decomposed into three 3-factors. If Tutte's 3-flow conjecture is true, it also holds for all 9-regular graphs with odd edge-connectivity 5, but not with odd edge-connectivity 3. It holds for all planar 2-edge-connected 9-regular graphs, an equivalent version of the 4-color theorem for planar graphs. We address the more general question: If G is an r-regular graph, and r=kq where k,q are natural numbers >1, can G be edge-decomposed into k q-factors? If q is even, then the decomposition exists trivially. If k,q are both odd, then we prove that the decomposition exists if G has odd edge-connectivity (size of smallest odd edge-cut) at least 3k−2, which is satisfied if the odd edge-connectivity is at least r−2. If q is odd and k is even, then we must require that G has an even number of vertices just to guarantee that G has a q-factor. If we want a decomposition into q-factors, then we also need the condition that, for any partition of the vertex set of G into two odd parts, there must be at least k edges between the parts. We prove that the edge-decomposition into q-factors is always possible if G has an even number of vertices and the edge-connectivity of G is at least 2k2+k.

We prove that every locally Hamiltonian graph with $n$ vertices and possibly with multiple edges has at least $3n-6$ edges with equality if and only if it triangulates the sphere. As a consequence, every edge-maximal embedding of a graph $G$ on some 2-dimensional surface $\Sigma$ (not necessarily compact) has at least $3n-6$ edges with equality if and only if $G$ also triangulates the sphere. If, in addition, $G$ is simple, then for each vertex $v$, the cyclic ordering of the edges around $v$ on $\Sigma$ is the same as the clockwise or anti-clockwise orientation around $v$ on the sphere. If $G$ contains no complete graph on 4 vertices, then the face-boundaries are the same in the two embeddings.

A well-known result of Tutte says that if $\Gamma$ is an Abelian group and $G$ is a graph having a nowhere-zero $\Gamma$-flow, then $G$ has a nowhere-zero $\Gamma'$-flow for each Abelian group $\Gamma'$ whose order is at least the order of $\Gamma$. Jaeger, Linial, Payan, and Tarsi observed that this does not extend to their more general concept of group connectivity. Motivated by this we define $g(k)$ as the least number such that, if $G$ is $\Gamma$-connected for some Abelian group $\Gamma$ of order $k$, then $G$ is also $\Gamma'$-connected for every Abelian group $\Gamma'$ of order $|\Gamma'| \geqslant g(k)$. We prove that $g(k)$ exists and satisfies for infinitely many $k$, \begin{align*}(2-o(1)) k < g(k) \leqslant 8k^3+1.\end{align*} The upper bound holds for all $k$. Analogously, we define $h(k)$ as the least number such that, if $G$ is $\Gamma$-colorable for some Abelian group $\Gamma$ of order $k$, then $G$ is also $\Gamma'$-colorable for every Abelian group $\Gamma'$ of order $|\Gamma'| \geq h(k)$. Then $h(k)$ exists and satisfies for infinitely many $k$, \begin{align*}(2-o(1)) k < h(k) < (2+o(1))k \ln(k).\end{align*} The upper bound (for all $k$) follows from a result of Král', Pangrác, and Voss. The lower bound follows by duality from our lower bound on $g(k)$ as that bound is demonstrated by planar graphs.

We show that in 5-connected planar and projective planar triangulations on n vertices, the number of Hamiltonian cycles grows exponentially with n. The result is best possible in the sense that 4-connected triangulations on n vertices on any fixed surface may have only polynomially many cycles. Also, there is an infinite class of 5-connected graphs (not on a fixed surface) which have only polynomially many cycles. The result also extends to 5-connected triangulations of the torus if a long standing conjecture of Nash-Williams holds. For any fixed surface, we show that every 5-connected triangulation of large face-width on that surface contains exponentially many cycles.

A graph G is locally connected if for every v∈V(G) the open neighbourhood N(v) of v is nonempty and induces a connected graph in G. We characterize locally connected graphs of order n with less than 2n edges and show that for any natural number k the Hamilton Cycle Problem for locally connected graphs of order n with m edges is polynomially solvable if m≤2n+klog2n, but NP-complete if m=2n+⌊n1∕k⌋.