Abstract. Consider a graph [Formula: see text] with a long path [Formula: see text]. When is it the case that [Formula: see text] also contains a long induced path? This question has been investigated in general as well as within a number of different graph classes since the 1980s. We have recently observed in a companion paper [ Long induced paths in sparse graphs and graphs with forbidden patterns, preprint, arXiv:2411.08685, 2024] that most existing results can be recovered in a simple way by considering forbidden ordered patterns of edges along the path [Formula: see text]. In particular, we proved that if we forbid some fixed ordered matching along a path of order [Formula: see text] in a graph [Formula: see text], then [Formula: see text] must contain an induced path of order [Formula: see text]. Moreover, we completely characterized the forbidden ordered patterns forcing the existence of an induced path of polynomial size. The purpose of the present paper is to completely characterize the ordered patterns [Formula: see text] such that forbidding [Formula: see text] along a path [Formula: see text] of order [Formula: see text] implies the existence of an induced path of order [Formula: see text]. These patterns are star forests with some specific ordering, which we call constellations. As a direct consequence of our result, we show that if a graph [Formula: see text] has a path of length [Formula: see text] and does not contain [Formula: see text] as a topological minor, then [Formula: see text] contains an induced path of order [Formula: see text]. The previously best known bound was [Formula: see text] for some unspecified function [Formula: see text] depending on the Topological Minor Structure Theorem of Grohe and Marx (2015).

The goal of local certification is to locally convince the vertices of a graph $G$ that $G$ satisfies a given property. A prover assigns short certificates to the vertices of the graph, then the vertices are allowed to check their certificates and the certificates of their neighbors, and based only on this local view, they must decide whether $G$ satisfies the given property. If the graph indeed satisfies the property, all vertices must accept the instance, and otherwise at least one vertex must reject the instance (for any possible assignment of certificates). The goal is to minimize the size of the certificates. In this paper we study the local certification of geometric and topological graph classes. While it is known that in $n$-vertex graphs, planarity can be certified locally with certificates of size $O(\log n)$, we show that several closely related graph classes require certificates of size $Ω(n)$. This includes penny graphs, unit-distance graphs, (induced) subgraphs of the square grid, 1-planar graphs, and unit-square graphs. These bounds are tight up to a constant factor and give the first known examples of hereditary (and even monotone) graph classes for which the certificates must have linear size. For unit-disk graphs we obtain a lower bound of $Ω(n^{1-δ})$ for any $δ>0$ on the size of the certificates, and an upper bound of $O(n \log n)$. The lower bounds are obtained by proving rigidity properties of the considered graphs, which might be of independent interest.

Abstract. For any hereditary graph class [Formula: see text], we construct optimal adjacency labeling schemes for the classes of subgraphs and induced subgraphs of Cartesian products of graphs in [Formula: see text]. As a consequence, we show that if [Formula: see text] admits efficient adjacency labels (or, equivalently, small induced-universal graphs) meeting the information-theoretic minimum, then so do the classes of subgraphs and induced subgraphs of Cartesian products of graphs in [Formula: see text]. Our proof uses ideas from randomized communication complexity, hashing, and additive combinatorics and improves upon recent results of Chepoi, Labourel, and Ratel [ J. Graph Theory, 93 (2020), pp. 64–87].

Hadwiger’s Conjecture asserts that every $K_{h}$-minor-free graph is properly $(h-1)$-colourable. We prove the following improper analogue of Hadwiger’s Conjecture: for fixed h, every $K_{h}$-minor-free graph is $(h-1)$-colourable with monochromatic components of bounded size. The number of colours is best possible regardless of the size of monochromatic components. It solves an open problem of Edwards, Kang, Kim, Oum and Seymour [SIAM J. Disc. Math. 2015], and concludes a line of research initiated in 2007. Similarly, for fixed $t \geqslant s$, we show that every $K_{s, t}$-minor-free graph is $(s+1)$-colourable with monochromatic components of bounded size. The number of colours is best possible, solving an open problem of van den Heuvel and Wood [J. London Math. Soc. 2018]. We actually prove a single theorem from which both of the above results are immediate corollaries. For an excluded apex minor, we strengthen the result as follows: for fixed $t \geqslant s \geqslant 3$, and for any fixed apex graph X, every $K_{s, t}$-subgraph-free X-minor-free graph is $(s+1)$-colourable with monochromatic components of bounded size. The number of colours is again best possible.

For any hereditary graph class $F$, we construct optimal adjacency labeling schemes for the classes of subgraphs and induced subgraphs of Cartesian products of graphs in $F$. As a consequence, we show that, if $F$ admits efficient adjacency labels (or, equivalently, small induced-universal graphs) meeting the information-theoretic minimum, then the classes of subgraphs and induced subgraphs of Cartesian products of graphs in $F$ do too. Our proof uses ideas from randomized communication complexity, hashing, and additive combinatorics, and improves upon recent results of Chepoi, Labourel, and Ratel [Journal of Graph Theory, 2020].

A graph is Ok-free if it does not contain k pairwise vertex-disjoint and non-adjacent cycles. We show that MAXIMUM INDEPENDENT SET and 3-COLORING in Ok-free graphs can be solved in quasi-polynomial time. As a main technical result, we establish that “sparse” (here, not containing large complete bipartite graphs as subgraphs) Ok-free graphs have treewidth (even, feedback vertex set number) at most logarithmic in the number of vertices. This is proven sharp as there is an infinite family of O2-free graphs without K3,3-subgraph and whose treewidth is (at least) logarithmic. Other consequences include that most of the central NP-complete problems (such as MAXIMUM INDEPENDENT SET, MINIMUM VERTEX COVER, MINIMUM DOMINATING SET, MINIMUM COLORING) can be solved in polynomial time in sparse Ok-free graphs, and that deciding the Ok-freeness of sparse graphs is polynomial time solvable. * This work was supported by the ANR projects DISTANCIA (ANR-17-CE40-0015), DIGRAPHS (ANR-19-CE48-0013-01), and TWIN-WIDTH (ANR-21-CE48-0014-01), by the LabEx PERSYVAL-lab (ANR-11-LABX-0025), and by the Vanier Canada Graduate Scholarships program. † The full version of the paper can be accessed at https://arxiv.org/abs/2206.00594

We study the two-player communication problem of determining whether two vertices $x, y$ are nearby in a graph $G$, with the goal of determining the graph structures that allow the problem to be solved with a constant-cost randomized protocol. Equivalently, we consider the problem of assigning constant-size random labels (sketches) to the vertices of a graph, which allow adjacency, exact distance thresholds, or approximate distance thresholds to be computed with high probability from the labels. Our main results are that, for monotone classes of graphs: constant-size adjacency sketches exist if and only if the class has bounded arboricity; constant-size sketches for exact distance thresholds exist if and only if the class has bounded expansion; constant-size approximate distance threshold (ADT) sketches imply that the class has bounded expansion; any class of constant expansion (i.e. any proper minor closed class) has constant-size ADT sketches; and a class may have arbitrarily small expansion without admitting constant-size ADT sketches.

The main problem in the area of graph property testing is to understand which graph properties are testable, which means that with constantly many queries to any input graph G, a tester can decide with good probability whether G satisfies the property, or is far from satisfying the property. Testable properties are well understood in the dense model and in the bounded degree model, but little is known in sparse graph classes when graphs are allowed to have unbounded degree. This is the setting of the sparse model. We prove that for any proper minor-closed class 𝒢, any monotone property (i.e., any property that is closed under taking subgraphs) is testable for graphs from 𝒢 in the sparse model. This extends a result of Czumaj and Sohler (FOCS'19), who proved it for monotone properties with finitely many forbidden subgraphs. Our result implies for instance that for any integers k and t, k-colorability of K_t-minor free graphs is testable in the sparse model. Elek recently proved that monotone properties of bounded degree graphs from minor-closed classes that are closed under disjoint union can be verified by an approximate proof labeling scheme in constant time. We show again that the assumption of bounded degree can be omitted in his result.

We construct asymptotically optimal adjacency labelling schemes for every hereditary class containing 2Ω(n2) n-vertex graphs as n→ ∞. This regime contains many classes of interest, for instance perfect graphs or comparability graphs, for which we obtain an adjacency labelling scheme with labels of n/4+o(n) bits per vertex. This implies the existence of a reachability labelling scheme for digraphs with labels of n/4+o(n) bits per vertex and comparability labelling scheme for posets with labels of n/4+o(n) bits per element. All these results are best possible, up to the lower order term.

We show that there exists an adjacency labelling scheme for planar graphs where each vertex of an n -vertex planar graph G is assigned a (1 + o(1)) log 2 n -bit label and the labels of two vertices u and v are sufficient to determine if uv is an edge of G . This is optimal up to the lower order term and is the first such asymptotically optimal result. An alternative, but equivalent, interpretation of this result is that, for every positive integer n , there exists a graph U n with n 1+o(1) vertices such that every n -vertex planar graph is an induced subgraph of U n . These results generalize to a number of other graph classes, including bounded genus graphs, apex-minor-free graphs, bounded-degree graphs from minor closed families, and k -planar graphs.

A subgraph $H$ of a graph $G$ is isometric if the distances between vertices in $H$ coincide with the distances between the corresponding vertices in $G$. We show that for any integer $n\ge 1$, there is a graph on $3^{n+O(\log^2 n)}$ vertices that contains isometric copies of all $n$-vertex graphs. Our main tool is a new type of distance labelling scheme, whose study might be of independent interest.

We show that there exists an adjacency labelling scheme for planar graphs where each vertex of an n-vertex planar graph G is assigned a (1+o(1))log2n-bit label and the labels of two vertices u and v are sufficient to determine if uv is an edge of G. This is optimal up to the lower order term and is the first such asymptotically optimal result. An alternative, but equivalent, interpretation of this result is that, for every positive integer n, there exists a graph Un with n1+o(1)vertices such that every n-vertex planar graph is an induced subgraph of Un. These results generalize to a number of other graph classes, including bounded genus graphs, apex-minor-free graphs, bounded-degree graphs from minor closed families, and k-planar graphs.

This paper studies sufficient conditions to obtain efficient distributed algorithms coloring graphs optimally (i.e.\ with the minimum number of colors) in the LOCAL model of computation. Most of the work on distributed vertex coloring so far has focused on coloring graphs of maximum degree $Δ$ with at most $Δ+1$ colors (or $Δ$ colors when some simple obstructions are forbidden). When $Δ$ is sufficiently large and $c\ge Δ-k_Δ+1$, for some integer $k_Δ\approx \sqrtΔ-2$, we give a distributed algorithm that given a $c$-colorable graph $G$ of maximum degree $Δ$, finds a $c$-coloring of $G$ in $\min\{O((\logΔ)^{1/12}\log n), 2^{O(\log Δ+\sqrt{\log \log n})}\}$ rounds, with high probability. The lower bound $Δ-k_Δ+1$ is best possible in the sense that for infinitely many values of $Δ$, we prove that when $χ(G)\le Δ-k_Δ$, finding an optimal coloring of $G$ requires $Ω(n)$ rounds. Our proof is a light adaptation of a remarkable result of Molloy and Reed, who proved that for $Δ$ large enough, for any $c\ge Δ- k_Δ$ deciding whether $χ(G)\le c$ is in {\textsf{P}}, while Embden-Weinert \emph{et al.}\ proved that for $c\le Δ-k_Δ-1$, the same problem is {\textsf{NP}}-complete. Note that the sequential and distributed thresholds differ by one. We also show that for any sufficiently large $Δ$, and $Ω(\log Δ)\le k \le Δ/100$, every graph of maximum degree $Δ$ and clique number at most $Δ-k$ can be efficiently colored with at most $Δ-\varepsilon k$ colors, for some absolute constant $\varepsilon >0$, with a randomized algorithm running in $O(\log n/\log \log n)$ rounds with high probability.

It was conjectured by Jaeger et al. in 1992 that for any prime number $p$, there is a constant $c$ such that for any $n$, the union (with repetition) of the vectors of any family of $c$ linear bases of $\mathbb{Z}_p^n$ forms an additive basis of $\mathbb{Z}_p^n$ (i.e., any element of $\mathbb{Z}_p^n$ can be expressed as the sum of a subset of these vectors). In this note, we prove this conjecture when each vector contains at most two nonzero entries. As an application, we prove several results on flows in highly edge-connected graphs, extending known results. For instance, assume that $p\geqslant 3$ is a prime number and $\vec{G}$ is a directed, highly edge-connected graph in which each arc is given a list of two distinct values in $\mathbb{Z}_p$. Then $\vec{G}$ has a $\mathbb{Z}_p$-flow in which each arc is assigned a value of its own list.

A Jordan region is a subset of the plane that is homeomorphic to a closed disk. Consider a family $\mathcal{F}$ of Jordan regions whose interiors are pairwise disjoint, and such that any two Jordan regions intersect in at most one point. If any point of the plane is contained in at most $k$ elements of $\mathcal{F}$ (with $k$ sufficiently large), then we show that the elements of $\mathcal{F}$ can be colored with at most k+1 colors so that intersecting Jordan regions are assigned distinct colors. This is best possible and answers a question raised by Reed and Shepherd in 1996. As a simple corollary, we also obtain a positive answer to a problem of Hlin\vený (1998) on the chromatic number of contact systems of strings. We also investigate the chromatic number of families of touching Jordan curves. This can be used to bound the ratio between the maximum number of vertex-disjoint directed cycles in a planar digraph, and its fractional counterpart.

An island in a graph is a set $X$ of vertices such that each element of $X$ has few neighbors outside $X$. In this paper, we prove several bounds on the size of islands in large graphs embeddable on fixed surfaces. As direct consequences of our results, we obtain the following: (1) Every graph of genus $g$ can be colored from lists of size 5, in such a way that each monochromatic component has size $O(g)$. Moreover, all but $O(g)$ vertices lie in monochromatic components of size at most 3. (2) Every triangle-free graph of genus $g$ can be colored from lists of size 3, in such a way that each monochromatic component has size $O(g)$. Moreover, all but $O(g)$ vertices lie in monochromatic components of size at most 10. (3) Every graph of girth at least 6 and genus $g$ can be colored from lists of size 2, in such a way that each monochromatic component has size $O(g)$. Moreover, all but $O(g)$ vertices lie in monochromatic components of size at most 16. While (2) is optimal up to the size of the components, we conjecture that the size of the lists can be decreased to 4 in (1), and the girth can be decreased to 5 in (3). We also study the complexity of minimizing the size of monochromatic components in 2-colorings of planar graphs.

We introduce the notion of (A, B)-colouring of a graph: For given vertex sets A, B, this is a colouring of the vertices in B so that both adjacent vertices and vertices with a common neighbour in A receive different colours. This concept generalises the notion of colouring the square of graphs and of cyclic colouring of plane graphs. We prove a general result which implies asymptotic versions of Wegner's and Borodin's Conjecture on these two colourings. Using a recent approach of Havet et al., we reduce the problem to edge-colouring of multigraphs and then use Kahn's result that the list chromatic index is close from the fractional chromatic index. Our results are based on a strong structural lemma for planar graphs which also implies that the size of a clique in the square of a planar graph of maximum degree Δ is at most Δ plus a constant.