A partial orientation $\vec{H}$ of a graph $G$ is a weak $r$-guidance system if for any two vertices at distance at most $r$ in $G$, there exists a shortest path $P$ between them such that $\vec{H}$ directs all but one edge in $P$ towards this edge. In case $\vec{H}$ has bounded maximum outdegree, this gives an efficient representation of shortest paths of length at most $r$ in $G$. We show that graphs from many natural graph classes admit such weak guidance systems, and study the algorithmic aspects of this notion.

Abstract. For an abelian group [Formula: see text], a graph [Formula: see text] is said to be [Formula: see text]-flow-critical if [Formula: see text] does not admit a nowhere-zero [Formula: see text]-flow, but for each edge [Formula: see text], the contraction [Formula: see text] has a nowhere-zero [Formula: see text]-flow. We obtain a bound on the density of [Formula: see text]-flow-critical graphs drawn on a fixed surface, generalizing the planar case of the bound on the density of 4-critical graphs by Kostochka and Yancey.

Two boxes in $\mathbb{R}^d$ are comparable if one of them is a subset of a translation of the other one. The comparable box dimension of a graph $G$ is the minimum integer $d$ such that $G$ can be represented as a touching graph of comparable axis-aligned boxes in $\mathbb{R}^d$. We show that proper minor-closed classes have bounded comparable box dimensions and explore further properties of this notion.

Weak and strong coloring numbers are generalizations of the degeneracy of a graph, where for each natural number $k$, we seek a vertex ordering such every vertex can (weakly respectively strongly) reach in $k$ steps only few vertices with lower index in the ordering. Both notions capture the sparsity of a graph or a graph class, and have interesting applications in the structural and algorithmic graph theory. Recently, the first author together with McCarty and Norin observed a natural volume-based upper bound for the strong coloring numbers of intersection graphs of well-behaved objects in $\mathbb{R}^d$, such as homothets of a centrally symmetric compact convex object, or comparable axis-aligned boxes. In this paper, we prove upper and lower bounds for the $k$-th weak coloring numbers of these classes of intersection graphs. As a consequence, we describe a natural graph class whose strong coloring numbers are polynomial in $k$, but the weak coloring numbers are exponential. We also observe a surprising difference in terms of the dependence of the weak coloring numbers on the dimension between touching graphs of balls (single-exponential) and hypercubes (double-exponential).

We give polynomial-time approximation schemes for monotone maximization problems expressible in terms of distances (up to a fixed upper bound) and efficiently solvable in graphs of bounded treewidth. These schemes apply in all fractionally treewidth-fragile graph classes, a property that is true for many natural graph classes with sublinear separators. We also provide quasipolynomial-time approximation schemes for these problems in all classes with sublinear separators.

We give a natural sufficient condition for an intersection graph of compact convex sets in $\mathbb{R}^d$ to have a balanced separator of sublinear size. This condition generalizes several previous results on sublinear separators in intersection graphs. Furthermore, the argument used to prove the existence of sublinear separators is based on a connection with generalized coloring numbers which has not been previously explored in geometric settings.

Baker [1] devised a technique to obtain approximation schemes for many optimization problems restricted to planar graphs; her technique was later extended to more general graph classes. In particular, using the Baker's technique and the minor structure theorem, Dawar et al. [5] gave Polynomial-Time Approximation Schemes (PTAS) for all monotone optimization problems expressible in the first-order logic when restricted to a proper minor-closed class of graphs. We define a Baker game formalizing the notion of repeated application of Baker's technique interspersed with vertex removal, prove that monotone optimization problems expressible in the first-order logic admit PTAS when restricted to graph classes in which the Baker game can be won in a constant number of rounds, and prove without use of the minor structure theorem that all proper minor-closed classes of graphs have this property.

A graph $G$ is $(a:b)$-colorable if there exists an assignment of $b$-element subsets of $\{1,\ldots,a\}$ to vertices of $G$ such that sets assigned to adjacent vertices are disjoint. We first show that for every triangle-free planar graph $G$ and a vertex $x\in V(G)$, the graph $G$ has a set coloring $\varphi$ by subsets of $\{1,\ldots,6\}$ such that $|\varphi(v)|\geq 2$ for $v\in V(G)$ and $|\varphi(x)|=3$. As a corollary, every triangle-free planar graph on $n$ vertices is $(6n:2n+1)$-colorable. We further use this result to prove that for every $\Delta$, there exists a constant $M_{\Delta}$ such that every planar graph $G$ of girth at least five and maximum degree $\Delta$ is $(6M_{\Delta}:2M_{\Delta}+1)$-colorable. Consequently, planar graphs of girth at least five with bounded maximum degree $\Delta$ have fractional chromatic number at most $3-\frac{3}{2M_{\Delta}+1}$.

We study $c$-crossing-critical graphs, which are the minimal graphs that require at least $c$ edge-crossings when drawn in the plane. For every fixed pair of integers with $c\ge 13$ and $d\ge 1$, we give first explicit constructions of $c$-crossing-critical graphs containing a vertex of degree greater than $d$. We also show that such unbounded degree constructions do not exist for $c\le 12$, precisely, that there exists a constant $D$ such that every $c$-crossing-critical graph with $c\le 12$ has maximum degree at most $D$. Hence, the bounded maximum degree conjecture of $c$-crossing-critical graphs, which was generally disproved in 2010 by Dvořák and Mohar (without an explicit construction), holds true, surprisingly, exactly for the values $c\le 12.$

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.

Baker [1] devised a powerful technique to obtain approximation schemes for various problems restricted to planar graphs. Her technique can be directly extended to various other graph classes, among the most general ones the graphs avoiding a fixed apex graph as a minor. Further generalizations (e.g., to all proper minor closed graph classes) are known, but they use a combination of techniques and usually focus on somewhat restricted classes of problems. We present a new type of graph decompositions (thin systems of overlays) generalizing Baker's technique and leading to straightforward polynomial-time approximation schemes. We also show that many graph classes (all proper minor-closed classes, and all subgraph-closed classes with bounded maximum degree and strongly sublinear separators) admit such decompositions.

Dvořák, Král', and Thomas [ Three-Coloring Triangle-Free Graphs on Surfaces IV. Bounding Face Sizes of 4-Critical Graphs, preprint, arXiv:1404.6356v3, 2015; Three-Coloring Triangle-Free Graphs on Surfaces VI. 3-Colorability of Quadrangulations, preprint, arXiv:1509.01013, 2015] gave a description of the structure of triangle-free graphs on surfaces with respect to 3-coloring. Their description, however, contains two substructures (both related to graphs embedded in a plane with two precolored cycles) whose coloring properties are not entirely determined. In this paper, we fill these gaps.

Aksenov proved that in a planar graph $G$ with at most one triangle, every precoloring of a 4-cycle can be extended to a 3-coloring of $G$. We give an exact characterization of planar graphs with two triangles in which some precoloring of a 4-cycle does not extend. We apply this characterization to solve the precoloring extension problem from two 4-cycles in a triangle-free planar graph in the case that the precolored 4-cycles are separated by many disjoint 4-cycles. The latter result is used in follow-up papers [ SIAM J. Discrete Math., 31 (2017), pp. 865--874; SIAM J. Discrete Math., 32 (2018), pp. 94--105] to give detailed information about the structure of 4-critical triangle-free graphs embedded in a fixed surface.

We present a polynomial-time 9/7-approximation algorithm for the graphic TSP for cubic graphs, which improves the previously best approximation factor of 1.3 for 2-connected cubic graphs and drops the requirement of 2-connectivity at the same time. To design our algorithm, we prove that every simple 2-connected cubic n-vertex graph contains a spanning closed walk of length at most 9n/7-1, and that such a walk can be found in polynomial time.

By Brook's Theorem, every n-vertex graph of maximum degree at most Delta >= 3 and clique number at most Delta is Delta-colorable, and thus it has an independent set of size at least n/Delta. We give an approximate characterization of graphs with independence number close to this bound, and use it to show that the problem of deciding whether such a graph has an independent set of size at least n/Delta+k has a kernel of size O(k).

We study 3-coloring properties of triangle-free planar graphs $G$ with two precolored 4-cycles $C_1$ and $C_2$ that are far apart. We prove that either every precoloring of $C_1\cup C_2$ extends to a 3-coloring of $G$, or $G$ contains one of two special substructures which uniquely determine which 3-colorings of $C_1\cup C_2$ extend. As a corollary, we prove that there exists a constant $D>0$ such that if $H$ is a planar triangle-free graph and if $S\subseteq V(H)$ consists of vertices at pairwise distances at least $D$, then every precoloring of $S$ extends to a 3-coloring of $H$. This gives a positive answer to a conjecture of Dvořák, Král', and Thomas, and implies an exponential lower bound on the number of 3-colorings of triangle-free planar graphs of bounded maximum degree.

A result of Plotkin, Rao, and Smith implies that graphs with polynomial expansion have strongly sublinear separators. We prove a converse of this result showing that hereditary classes of graphs with strongly sublinear separators have polynomial expansion. This confirms a conjecture of the first author.

Every triangle-free planar graph on $n$ vertices has an independent set of size at least $(n+1)/3$, and this lower bound is tight. We give an algorithm that, given a triangle-free planar graph $G$ on $n$ vertices and an integer $k\geq0$, decides whether $G$ has an independent set of size at least $(n+k)/3$, in time $2^{O(\sqrt{k})}n$. Thus, the problem is fixed-parameter tractable when parameterized by $k$. Furthermore, as a corollary of the result used to prove the correctness of the algorithm, we show that there exists $\varepsilon>0$ such that every planar graph of girth at least five on $n$ vertices has an independent set of size at least $n/(3-\varepsilon)$. We further give an algorithm that, given a planar graph $G$ of maximum degree 4 on $n$ vertices and an integer $k\geq0$, decides whether $G$ has an independent set of size at least $(n+k)/4$, in time $2^{O(\sqrt{k})}n$.

A graph $H$ is strongly immersed in $G$ if $G$ is obtained from $H$ by a sequence of vertex splittings (i.e., lifting some pairs of incident edges and removing the vertex) and edge removals. Equivalently, vertices of $H$ are mapped to distinct vertices of $G$ (branch vertices), and edges of $H$ are mapped to pairwise edge-disjoint paths in $G$, each of them joining the branch vertices corresponding to the ends of the edge and not containing any other branch vertices. We show that there exists a function $d\colon N\to N$ such that for all graphs $H$ and $G$, if $G$ contains a strong immersion of the star $K_{1,d(\Delta(H))|V(H)|}$ whose branch vertices are $\Delta(H)$-edge-connected to one another, then $H$ is strongly immersed in $G$. This has a number of structural consequences for graphs avoiding a strong immersion of $H$. In particular, a class $\mathcal{G}$ of simple 4-edge-connected graphs contains all graphs of maximum degree 4 as strong immersions if and only if $\mathcal{G}$ has either unbounded maximum degree or unbounded tree-width.

We show that if $G$ is a 4-critical graph embedded in a fixed surface $\Sigma$ so that every contractible cycle has length at least 5, then $G$ can be expressed as $G=G'\cup G_1\cup G_2\cup\cdots\cup G_k$, where $|V(G')|$ and $k$ are bounded by a constant (depending linearly on the genus of $\Sigma$) and $G_1, \ldots, G_k$ are graphs (of unbounded size) whose structure we describe exactly. The proof is computer assisted---we use a computer to enumerate all plane 4-critical graphs of girth 5 with a precolored cycle of length at most 16 that are used in the basic case of the inductive proof of the statement.

For any fixed surface σ of genus g, we give an algorithm to decide whether a graph G of girth at least five embedded in σ is colorable from an assignment of lists of size three in time O(|V(G)|). Furthermore, we can allow a subgraph (of any size) with at most s components to be precolored, at the expense of increasing the time complexity of the algorithm to O(|V(G)|K(g+s)+1) for some absolute constant K; in both cases, the multiplicative constant hidden in the O-notation depends on g and s. This also enables us to find such a coloring when it exists. The idea of the algorithm can be applied to other similar problems, e.g., 5-list-coloring of graphs on surfaces.

We present a linear-time algorithm for deciding first-order (FO) properties in classes of graphs with bounded expansion, a notion recently introduced by Nešetřil and Ossona de Mendez. This generalizes several results from the literature, because many natural classes of graphs have bounded expansion: graphs of bounded tree-width, all proper minor-closed classes of graphs, graphs of bounded degree, graphs with no subgraph isomorphic to a subdivision of a fixed graph, and graphs that can be drawn in a fixed surface in such a way that each edge crosses at most a constant number of other edges. We deduce that there is an almost linear-time algorithm for deciding FO properties in classes of graphs with locally bounded expansion. More generally, we design a dynamic data structure for graphs belonging to a fixed class of graphs of bounded expansion. After a linear-time initialization the data structure allows us to test an FO property in constant time, and the data structure can be updated in constant time after addition/deletion of an edge, provided the list of possible edges to be added is known in advance and their simultaneous addition results in a graph in the class. All our results also hold for relational structures and are based on the seminal result of Nešetřil and Ossona de Mendez on the existence of low tree-depth colorings.

A graph G is k-choosable if G can be properly colored whenever every vertex has a list of at least k available colors. Thomassen's theorem states that every planar graph is 5-choosable. We extend the result by showing that every graph with at most two crossings is 5-choosable.

Grötzsch's theorem states that every triangle-free planar graph is 3-colorable, and several relatively simple proofs of this fact were provided by Thomassen and other authors. It is easy to convert these proofs into quadratic-time algorithms to find a 3-coloring, but it is not clear how to find such a coloring in linear time (Kowalik used a nontrivial data structure to construct anO(nlogn) algorithm). We design a linear-time algorithm to find a 3-coloring of a given triangle-free planar graph. The algorithm avoids using any complex data structures, which makes it easy to implement. As a by-product, we give a yet simpler proof of Grötzsch's theorem.

We present a linear-time algorithm for deciding first-order logic (FOL) properties in classes of graphs with bounded expansion. Many natural classes of graphs have bounded expansion: graphs of bounded tree-width, all proper minor-closed classes of graphs, graphs of bounded degree, graphs with no sub graph isomorphic to a subdivision of a fixed graph, and graphs that can be drawn in a fixed surface in such a way that each edge crosses at most a constant number of other edges. We also develop an almost linear-time algorithm for deciding FOL properties in classes of graphs with locally bounded expansion, those include classes of graphs with locally bounded tree-width or locally excluding a minor. More generally, we design a dynamic data structure for graphs belonging to a fixed class of graphs of bounded expansion. After a linear-time initialization the data structure allows us to test an FOL property in constant time, and the data structure can be updated in constant time after addition/deletion of an edge, provided the list of possible edges to be added is known in advance and their addition results in a graph in the class. In addition, we design a dynamic data structure for testing existential properties or the existence of short paths between prescribed vertices in such classes of graphs. All our results also hold for relational structures and are based on the seminal result of Nesetril and Ossona de Mendez on the existence of low tree-depth colorings.

A graph is k-choosable if it can be colored whenever every vertex has a list of at least k available colors. We prove that if a triangle-free planar graph is not 3-choosable, then it contains a 4-cycle that intersects another 4- or 5-cycle in exactly one edge. This strengthens Thomassen's result [C. Thomassen, J. Combin. Theory Ser. B, 64 (1995), pp. 101–107] that every planar graph of girth at least 5 is 3-choosable. In addition, this implies that every triangle-free planar graph without 6- and 7-cycles is 3-choosable.

Gimbel and Thomassen asked whether 3-colorability of a triangle-free graph drawn on a fixed surface can be tested in polynomial time. We settle the question by giving a linear-time algorithm for every surface which combined with previous results gives a linear-time algorithm to compute the chromatic number of such graphs. Our algorithm is based on a structure theorem that for a triangle-free graph drawn on a surface Σ guarantees the existence of a subgraph H, whose size depends only on Σ, such that there is an easy test whether a 3-coloring of H extends to a 3-coloring of G. The test is based on a topological obstruction, called the “winding number” of a 3-coloring. To prove the structure theorem we make use of disjoint paths with specified ends to find a 3-coloring. If the input triangle-free graph G drawn in Σ is 3-colorable we can find a 3-coloring in quadratic time, and if G quadrangulates Σ then we can find the 3-coloring in linear time. The latter algorithm requires two ingredients that may be of independent interest: a generalization of a data structure of Kowalik and Kurowski to weighted graphs and a speedup of a disjoint paths algorithm of Robertson and Seymour to linear time.

Grötzsch's theorem states that every triangle-free planar graph is 3-colorable, and several relatively simple proofs of this fact were provided by Thomassen and other authors. It is easy to convert these proofs into quadratic-time algorithms to find a 3-coloring, but it is not clear how to find such a coloring in linear time (Kowalik used a nontrivial data structure to construct an O(n log n) algorithm). We design a linear-time algorithm to find a 3-coloring of a given triangle-free planar graph. The algorithm avoids using any complex data structures, which makes it easy to implement. As a by-product we give a yet simpler proof of Grötzsch's theorem.

We show that the maximum number of colors that can be used in a vertex coloring of a cubic 3-connected plane graph G that avoids a face with vertices of mutually distinct colors (a rainbow face) is equal to $\frac{n}{2}+\mu^*-2$, where n is the number of vertices of G and $\mu^*$ is the size of the maximum matching of the dual graph $G^*$.

A well-known result (Heawood [Quart. J. Pure Appl. Math., 24 (1890), pp. 332–338], Ringel [Map Color Theorem, Springer-Verlag, New York, 1974], Ringel and Youngs [Proc. Nat. Acad. Sci., U.S.A., 60 (1968), pp. 438–445]) states that the maximum chromatic number of a graph embedded in a given surface S coincides with the size of the largest clique that can be embedded in S, and that this number can be expressed as a simple formula in the Euler genus of S. A partition of a graph G into k parts consists of k edge-disjoint subgraphs $G_1,\dots,G_k$ such that $E(G)=E(G_1)\cup E(G_2)\cup\dots\cup E(G_k)$. The k-chromatic number $\chi_k (G)$ is the maximum of $\sum_{i=1}^k\chi(G_i)$ over all partitions of G into k parts. We derive a Heawood-type formula for the k-chromatic number of graphs embedded in a fixed surface, improving the previously known upper bounds. In infinitely many cases, the new upper bound coincides with the lower bound obtained from embedding disjoint cliques in the surface. In the proof of this result, we derive a variant of Euler's formula for the union of several graphs that might be interesting independently.

The problem of coloring the square of a graph naturally arises in connection with the distance labelings, which have been studied intensively. We consider this problem for sparse subcubic graphs. We show that the choosability $\chi_\ell(G^2)$ of the square of a subcubic graph G of maximum average degree d is at most four if $d<24/11$ and G does not contain a 5-cycle, at most five if $d<7/3$, and at most six if $d<5/2$. Wegner's conjecture claims that the chromatic number of the square of a subcubic planar graph is at most seven. Let G be a planar subcubic graph of girth g. Our result implies that $\chi_\ell(G^2)$ is at most four if $g\ge 24$, at most 5 if $g\ge 14$, and at most 6 if $g\ge 10$. For lower bounds, we find a planar subcubic graph $G_1$ of girth 9 such that $\chi(G_1^2)=5$ and a planar subcubic graph $G_2$ of girth 5 such that $\chi(G_2^2)=6$. As a consequence, we show that the problem of 4-coloring of the square of a subcubic planar graph of girth $g=9$ is NP-complete. We conclude the paper by posing a few conjectures.

Let G be a graph and let $c: V(G)\to\binom{1,\ldots,5}{2}$ be an assignment of 2-element subsets of the set $1,\ldots,5$ to the vertices of G such that for every edge $vw$, the sets $c(v)$ and $c(w)$ are disjoint. We call such an assignment a $(5,2)$-coloring. A graph is (5,2)-colorable if and only if it has a homomorphism to the Petersen graph. The odd-girth of a graph G is the length of the shortest odd cycle in G ($\infty$ if G is bipartite). We prove that every planar graph of odd-girth at least 9 is $(5,2)$-colorable, and thus it is homomorphic to the Petersen graph. Also, this implies that such graphs have a fractional chromatic number at most $5\over2$. As a special case, this result holds for planar graphs of girth at least 8.

In 1955 Kotzig [A. Kotzig, Math. Slovaca, 5 (1955), pp. 111-113] proved that every planar 3-connected graph contains an edge such that the sum of degrees of its end-vertices is at most $13$. Moreover, if the graph does not contain 3-vertices, then this sum is at most $11$. Such an edge is called light. The well-known result of Steinitz [E. Steinitz, Enzykl. Math. Wiss., 3 (1922), pp. 1-139] that the 3-connected planar graphs are precisely the skeletons of 3-polytopes gives an additional trump to Kotzig's theorem. On the other hand, in 1961, Tutte [W. T. Tutte, Indag. Math., 23 (1961), pp. 441-455] proved that every 3-connected graph, distinct from $K_4$, contains a contractible edge. In this paper, we strengthen Kotzig's theorem by showing that every 3-connected planar graph distinct from $K_4$ contains an edge that is both light and contractible. A consequence is that every 3-polytope can be constructed from tetrahedron by a sequence of splittings of vertices of degree at most $11$.