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For a fixed graph H, the H-Recoloring problem asks whether for two given homomorphisms from a graph G to H, we can transform one into the other by changing the image of a single vertex of G in each step and maintaining a homomorphism from G to H throughout. We extend an algorithm of Wrochna for H-Recoloring where H is a square-free loopless undirected graph to the more general setting of directed graphs. We obtain a polynomial-time algorithm for H-Recoloring in this setting whenever H is a loopless digraph that does not contain a 4-cycle of algebraic girth zero and whenever H is a reflexive digraph that contains neither a 3-cycle of algebraic girth 1 nor a 4-cycle of algebraic girth zero.

A dominating set in a directed graph is a set of vertices $S$ such that all the vertices that do not belong to $S$ have an in-neighbour in $S$. A locating set $S$ is a set of vertices such that all the vertices that do not belong to $S$ are characterized uniquely by the in-neighbours they have in $S$, i.e. for every two vertices $u$ and $v$ that are not in $S$, there exists a vertex $s\in S$ that dominates exactly one of them. The size of a smallest set of a directed graph $D$ which is both locating and dominating is denoted by $\gamma^{LD}(D)$. Foucaud, Heydarshahi and Parreau proved that any twin-free digraph $D$ satisfies $\gamma^{LD}(D)\leq \frac{4n} 5 +1$ but conjectured that this bound can be lowered to $\frac{2n} 3$. The conjecture is still open. They also proved that if $D$ is a tournament, i.e. a directed graph where there is one arc between every pair of vertices, then $\gamma^{LD}(D)\leq \lceil \frac{n}{2}\rceil$. The main result of this paper is the generalization of this bound to connected local tournaments, i.e. connected digraphs where the in- and out-neighbourhoods of every vertex induce a tournament. We also prove $\gamma^{LD}(D)\leq \frac{2n} 3$ for all quasi-twin-free digraphs $D$ that admit a supervising vertex (a vertex from which any vertex is reachable). This class of digraphs generalizes twin-free acyclic graphs, the most general class for which this bound was known.

A proof labelling scheme for a graph class $\mathcal{C}$ is an assignment of certificates to the vertices of any graph in the class $\mathcal{C}$, such that upon reading its certificate and the certificates of its neighbors, every vertex from a graph $G\in \mathcal{C}$ accepts the instance, while if $G\not\in \mathcal{C}$, for every possible assignment of certificates, at least one vertex rejects the instance. It was proved recently that for any fixed surface $\Sigma$, the class of graphs embeddable in $\Sigma$ has a proof labelling scheme in which each vertex of an $n$-vertex graph receives a certificate of at most $O(\log n)$ bits. The proof is quite long and intricate and heavily relies on an earlier result for planar graphs. Here we give a very short proof for any surface. The main idea is to encode a rotation system locally, together with a spanning tree supporting the local computation of the genus via Euler's formula., Comment: 10 pages, 5 figures - v4: final version

We prove that every planar graph is the intersection graph of homothetic triangles in the plane., Comment: 6 pages

We present a bijection for toroidal maps that are essentially $3$-connected ($3$-connected in the periodic planar representation). Our construction actually proceeds on certain closely related bipartite toroidal maps with all faces of degree $4$ except for a hexagonal root-face. We show that these maps are in bijection with certain well-characterized bipartite unicellular maps. Our bijection, closely related to the recent one by Bonichon and L\'ev\^eque for essentially 4-connected toroidal triangulations, can be seen as the toroidal counterpart of the one developed in the planar case by Fusy, Poulalhon and Schaeffer, and it extends the one recently proposed by Fusy and L\'ev\^eque for essentially simple toroidal triangulations. Moreover, we show that rooted essentially $3$-connected toroidal maps can be decomposed into two pieces, a toroidal part that is treated by our bijection, and a planar part that is treated by the above-mentioned planar case bijection. This yields a combinatorial derivation for the bivariate generating function of rooted essentially $3$-connected toroidal maps, counted by vertices and faces., Comment: 30 pages

We study the scaling limit of essentially simple triangulations on the torus. We consider, for every $n\geq 1$, a uniformly random triangulation $G_n$ over the set of (appropriately rooted) essentially simple triangulations on the torus with $n$ vertices. We view $G_n$ as a metric space by endowing its set of vertices with the graph distance denoted by $d_{G_n}$ and show that the random metric space $(V(G_n),n^{-1/4}d_{G_n})$ converges in distribution in the Gromov-Hausdorff sense when $n$ goes to infinity, at least along subsequences, toward a random metric space. One of the crucial steps in the argument is to construct a simple labeling on the map and show its convergence to an explicit scaling limit. We moreover show that this labeling approximates the distance to the root up to a uniform correction of order $o(n^{1/4})$., Comment: 93 pages

We present unified bijections for maps on the torus with control on the face-degrees and essential girth (girth of the periodic planar representation). A first step is to show that for d>=3 every toroidal d-angulation of essential girth d can be endowed with a certain "canonical" orientation (formulated as a weight-assignment on the half-edges). Using an adaptation of a construction by Bernardi and Chapuy, we can then derive a bijection between face-rooted toroidal d-angulations of essential girth d (with the condition that, apart from the root-face contour, no other closed walk of length d encloses the root-face) and a family of decorated unicellular maps. The orientations and bijections can then be generalized, for any d>=1, to toroidal face-rooted maps of essential girth d with a root-face of degree d (and with the same root-face contour condition as for d-angulations), and they take a simpler form in the bipartite case, as a parity specialization. On the enumerative side we obtain explicit algebraic expressions for the generating functions of rooted essentially simple triangulations and bipartite quadrangulations on the torus. Our bijective constructions can be considered as toroidal counterparts of those obtained by Bernardi and the first author in the planar case, and they also build on ideas introduced by Despr\'e, Gon\c{c}alves and the second author for essentially simple triangulations, of imposing a balancedness condition on the orientations in genus 1., Comment: 69 pages

Transversal structures (also known as regular edge labelings) are combinatorial structures defined over 4-connected plane triangulations with quadrangular outer-face. They have been intensively studied and used for many applications (drawing algorithm, random generation, enumeration ...). In this paper we introduce and study a generalization of these objects for the toroidal case. Contrary to what happens in the plane, the set of toroidal transversal structures of a given toroidal triangulation is partitioned into several distributive lattices. We exhibit a subset of toroidal transversal structures, called balanced, and show that it forms a single distributive lattice. Then, using the minimal element of the lattice, we are able to enumerate bijectively essentially 4-connected toroidal triangulations., Comment: 67 pages. arXiv admin note: text overlap with arXiv:1702.07589

Schnyder woods are particularly elegant combinatorial structures with numerous applications concerning planar triangulations and more generally 3-connected planar maps. We propose a simple generalization of Schnyder woods from the plane to maps on orientable surfaces of any genus with a special emphasis on the toroidal case. We provide a natural partition of the set of Schnyder woods of a given map into distributive lattices depending on the surface homology. In the toroidal case we show the existence of particular Schnyder woods with some global properties that are useful for optimal encoding or graph drawing purpose., Comment: 200 pages, Habilitation manuscript

Poulalhon and Schaeffer introduced an elegant method to linearly encode a planar triangulation optimally. The method is based on performing a special depth-first search algorithm on a particular orientation of the triangulation: the minimal Schnyder wood. Recent progress toward generalizing Schnyder woods to higher genus enables us to generalize this method to the toroidal case. In the plane, the method leads to a bijection between planar triangulations and some particular trees. For the torus we obtain a similar bijection but with particular unicellular maps (maps with only one face)., Comment: 41 pages

We propose a simple generalization of Schnyder woods from the plane to maps on orientable surfaces of higher genus. This is done in the language of angle labelings. Generalizing results of De Fraysseix and Ossona de Mendez, and Felsner, we establish a correspondence between these labelings and orientations and characterize the set of orientations of a map that correspond to such a Schnyder labeling. Furthermore, we study the set of these orientations of a given map and provide a natural partition into distributive lattices depending on the surface homology. This generalizes earlier results of Felsner and Ossona de Mendez. In the toroidal case, a new proof for the existence of Schnyder woods is derived from this approach.

For planar graphs, we consider the problems of \emph{list edge coloring} and \emph{list total coloring}. Edge coloring is the problem of coloring the edges while ensuring that two edges that are adjacent receive different colors. Total coloring is the problem of coloring the edges and the vertices while ensuring that two edges that are adjacent, two vertices that are adjacent, or a vertex and an edge that are incident receive different colors. In their list extensions, instead of having the same set of colors for the whole graph, every vertex or edge is assigned some set of colors and has to be colored from it. A graph is minimally edge or total choosable if it is list edge $\Delta$-colorable or list total $(\Delta+1)$-colorable, respectively, where $\Delta$ is the maximum degree in the graph. It is already known that planar graphs with $\Delta\geq 8$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable (Li Xu 2011), and that planar graphs with $\Delta\geq 7$ and no triangle sharing a vertex with a $C_4$ or no triangle adjacent to a $C_k$ ($\forall 3 \leq k \leq 6$) are minimally total colorable (Wang Wu 2011). We strengthen here these results and prove that planar graphs with $\Delta\geq 7$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable., Comment: 12 pages, 3 figures

We prove a decomposition theorem for graphs that do not contain a subdivision of $K_4$ as an induced subgraph where $K_4$ is the complete graph on four vertices. We obtain also a structure theorem for the class $\cal C$ of graphs that contain neither a subdivision of $K_4$ nor a wheel as an induced subgraph, where a wheel is a cycle on at least four vertices together with a vertex that has at least three neighbors on the cycle. Our structure theorem is used to prove that every graph in $\cal C$ is 3-colorable and entails a polynomial-time recognition algorithm for membership in $\cal C$. As an intermediate result, we prove a structure theorem for the graphs whose cycles are all chordless.

An \emph{s-graph} is a graph with two kinds of edges: \emph{subdivisible} edges and \emph{real} edges. A \emph{realisation} of an s-graph $B$ is any graph obtained by subdividing subdivisible edges of $B$ into paths of arbitrary length (at least one). Given an s-graph $B$, we study the decision problem $\Pi_B$ whose instance is a graph $G$ and question is "Does $G$ contain a realisation of $B$ as an induced subgraph?". For several $B$'s, the complexity of $\Pi_B$ is known and here we give the complexity for several more. Our NP-completeness proofs for $\Pi_B$'s rely on the NP-completeness proof of the following problem. Let $\cal S$ be a set of graphs and $d$ be an integer. Let $\Gamma_{\cal S}^d$ be the problem whose instance is $(G, x, y)$ where $G$ is a graph whose maximum degree is at most d, with no induced subgraph in $\cal S$ and $x, y \in V(G)$ are two non-adjacent vertices of degree 2. The question is "Does $G$ contain an induced cycle passing through $x, y$?". Among several results, we prove that $\Gamma^3_{\emptyset}$ is NP-complete. We give a simple criterion on a connected graph $H$ to decide whether $\Gamma^{+\infty}_{\{H\}}$ is polynomial or NP-complete. The polynomial cases rely on the algorithm three-in-a-tree, due to Chudnovsky and Seymour., Comment: arXiv admin note: text overlap with arXiv:1308.6678

We consider the problem of coloring the squares of graphs of bounded maximum average degree, that is, the problem of coloring the vertices while ensuring that two vertices that are adjacent or have a common neighbour receive different colors. Borodin et al. proved in 2004 and 2008 that the squares of planar graphs of girth at least seven and sufficiently large maximum degree $\Delta$ are list $(\Delta+1)$-colorable, while the squares of some planar graphs of girth six and arbitrarily large maximum degree are not. By Euler's Formula, planar graphs of girth at least $6$ are of maximum average degree less than $3$, and planar graphs of girth at least $7$ are of maximum average degree less than $14/5<3$. We strengthen their result and prove that there exists a function $f$ such that the square of any graph with maximum average degree $m<3$ and maximum degree $\Delta\geq f(m)$ is list $(\Delta+1)$-colorable. This bound of $3$ is optimal in the sense that the above-mentioned planar graphs with girth $6$ have maximum average degree less than $3$ and arbitrarily large maximum degree, while their square cannot be $(\Delta+1)$-colored. The same holds for list injective $\Delta$-coloring., Comment: 12 pages, 4 figures, submitted

For graphs of bounded maximum average degree, we consider the problem of 2-distance coloring. This is the problem of coloring the vertices while ensuring that two vertices that are adjacent or have a common neighbor receive different colors. It is already known that planar graphs of girth at least 6 and of maximum degree D are list 2-distance (D+2)-colorable when D>=24 (Borodin and Ivanova (2009)) and 2-distance (D+2)-colorable when D>=18 (Borodin and Ivanova (2009)). We prove here that D>=17 suffices in both cases. More generally, we show that graphs with maximum average degree less than 3 and D>=17 are list 2-distance (D+2)-colorable. The proof can be transposed to list injective (D+1)-coloring., Comment: 22 pages, 5 figures, submitted

We describe a tabulation of (conjecturally) modular elliptic curves over the field Q(sqrt(5)) up to the first curve of rank 2. Using an efficient implementation of an algorithm of Lassina Dembele, we computed tables of Hilbert modular forms of weight (2,2) over Q(sqrt(5)), and via a variety of methods we constructed corresponding elliptic curves, including (again, conjecturally) all elliptic curves over Q(sqrt(5)) that have conductor with norm less than or equal to 1831., Comment: 17 pages

A Schnyder wood is an orientation and coloring of the edges of a planar map satisfying a simple local property. We propose a generalization of Schnyder woods to graphs embedded on the torus with application to graph drawing. We prove several properties on this new object. Among all we prove that a graph embedded on the torus admits such a Schnyder wood if and only if it is an essentially 3-connected toroidal map. We show that these Schnyder woods can be used to embed the universal cover of an essentially 3-connected toroidal map on an infinite and periodic orthogonal surface. Finally we use this embedding to obtain a straight-line flat torus representation of any toroidal map in a polynomial size grid.

We analyse the origin of the Adler anomaly of chiral U(1) gauge theory within the framework of regularized path integrals. Momentum or position space regulators allow for mathematically well-defined path integrals but violate local gauge symmetry. It is known how (nonanomalous) gauge symmetry can be recovered in the renormalized theory in this case [1]. Here we analyse U(1) chiral gauge theory to show how the appearance of anomalies manifests itself in such a context. We show that the three-photon amplitude leads to a violation of the Slavnov-Taylor-Identities which cannot be restored on taking the UV limit in the renormalized theory. We point out that this fact is related to the nonanalyticity of this amplitude in the infrared region., Comment: 33 pages, 1 figure; few minor changes, 2 references added

Vertex deletion and edge deletion problems play a central role in Parameterized Complexity. Examples include classical problems like Feedback Vertex Set, Odd Cycle Transversal, and Chordal Deletion. Interestingly, the study of edge contraction problems of this type from a parameterized perspective has so far been left largely unexplored. We consider two basic edge contraction problems, which we call Path-Contractibility and Tree-Contractibility. Both problems take an undirected graph $G$ and an integer $k$ as input, and the task is to determine whether we can obtain a path or an acyclic graph, respectively, by contracting at most $k$ edges of $G$. Our main contribution is an algorithm with running time $4^{k+O(\log^2 k)} + n^{O(1)}$ for Path-Contractibility and an algorithm with running time $4.88^k n^{O(1)}$ for Tree-Contractibility, based on a novel application of the color coding technique of Alon, Yuster and Zwick. Furthermore, we show that Path-Contractibility has a kernel with at most $5k+3$ vertices, while Tree-Contractibility does not have a polynomial kernel unless coNP $\subseteq$ NP/poly. We find the latter result surprising, because of the strong connection between Tree-Contractibility and Feedback Vertex Set, which is known to have a vertex kernel with size $O(k^2)$.

An asteroidal triple is a stable set of three vertices such that each pair is connected by a path avoiding the neighborhood of the third vertex. Asteroidal triples play a central role in a classical characterization of interval graphs by Lekkerkerker and Boland. Their result says that a chordal graph is an interval graph if and only if it contains no asteroidal triple. In this paper, we prove an analogous theorem for directed path graphs which are the intersection graphs of directed paths in a directed tree. For this purpose, we introduce the notion of a strong path. Two non-adjacent vertices are linked by a strong path if either they have a common neighbor or they are the endpoints of two vertex-disjoint chordless paths satisfying certain conditions. A strong asteroidal triple is an asteroidal triple such that each pair is linked by a strong path. We prove that a chordal graph is a directed path graph if and only if it contains no strong asteroidal triple. We also introduce a related notion of asteroidal quadruple, and conjecture a characterization of rooted path graphs which are the intersection graphs of directed paths in a rooted tree.

A graph is a path graph if it is the intersection graph of a family of subpaths of a tree. In 1970, Renz asked for a characterizaton of path graphs by forbidden induced subgraphs. Here we answer this question by listing all graphs that are not path graphs and are minimal with this property.

We consider the problem of minimizing the size of a family of sets G such that every subset of 1,...,n can be written as a disjoint union of at most k members of G, where k and n are given numbers. This problem originates in a real-world application aiming at the diversity of industrial production. At the same time, the minimum of G so that every subset of 1,...,n is the union of two sets in G has been asked by Erdos and studied recently by Furedi and Katona without requiring the disjointness of the sets. A simple construction providing a feasible solution is conjectured to be optimal for this problem for all values of n and k and regardless of the disjointness requirement; we prove this conjecture in special cases including all (n,k) for which n <= 3k holds, and some individual values of n and k.

A Meyniel obstruction is an odd cycle with at least five vertices and at most one chord. A graph is Meyniel if and only if it has no Meyniel obstruction as an induced subgraph. Here we give a O(n^2) algorithm that, for any graph, finds either a clique and coloring of the same size or a Meyniel obstruction. We also give a O(n^3) algorithm that, for any graph, finds either aneasily recognizable strong stable set or a Meyniel obstruction.

The pre-coloring extension problem consists, given a graph $G$ and a subset of nodes to which some colors are already assigned, in finding a coloring of $G$ with the minimum number of colors which respects the pre-coloring assignment. This can be reduced to the usual coloring problem on a certain contracted graph. We prove that pre-coloring extension is polynomial for complements of Meyniel graphs. We answer a question of Hujter and Tuza by showing that ``PrExt perfect'' graphs are exactly the co-Meyniel graphs, which also generalizes results of Hujter and Tuza and of Hertz. Moreover we show that, given a co-Meyniel graph, the corresponding contracted graph belongs to a restricted class of perfect graphs (``co-Artemis'' graphs, which are ``co-perfectly contractile'' graphs), whose perfectness is easier to establish than the strong perfect graph theorem. However, the polynomiality of our algorithm still depends on the ellipsoid method for coloring perfect graphs.

We consider the class A of graphs that contain no odd hole, no antihole, and no ``prism'' (a graph consisting of two disjoint triangles with three disjoint paths between them). We show that the coloring algorithm found by the second and fourth author can be implemented in time O(n^2m) for any graph in A with n vertices and m edges, thereby improving on the complexity proposed in the original paper.

A Meyniel graph is a graph in which every odd cycle of length at least five has two chords. In the manuscript "Coloring Meyniel graphs in linear time" we claimed that our algorithm MCColor produces an optimal coloring for every Meyniel graph. But later we found a mistake in the proof and a couterexample to the optimality, which we present here. MCColor can still be used to find a stable set that intersects all maximal cliques of a Meyniel graph in linear time. Consequently it can be used to find an optimal coloring in time O(nm), and the same holds for Algorithm MCS+Color. This is explained in the manuscript "A linear algorithm to find a strong stable set in a Meyniel graph" but this is equivalent to Hertz's algorithm. The current best algorithm for coloring Meyniel graphs is the O(n^2) algorithm LexColor due to Roussel and Rusu. The question of finding a linear-time algorithm to color Meyniel graphs is still open.