Coloring decompositions of complete geometric graphs

A decomposition of a non-empty simple graph $G$ is a pair $[G,P]$, such that $P$ is a set of non-empty induced subgraphs of $G$, and every edge of $G$ belongs to exactly one subgraph in $P$. The chromatic index $\chi'([G,P])$ of a decomposition $[G,P]$ is the smallest number $k$ for which there exists a $k$-coloring of the elements of $P$ in such a way that: for every element of $P$ all of its edges have the same color, and if two members of $P$ share at least one vertex, then they have different colors. A long standing conjecture of Erd\H{o}s-Faber-Lov\'asz states that every decomposition $[K_n,P]$ of the complete graph $K_n$ satisfies $\chi'([K_n,P])\leq n$. In this paper we work with geometric graphs, and inspired by this formulation of the conjecture, we introduce the concept of chromatic index of a decomposition of the complete geometric graph. We present bounds for the chromatic index of several types of decompositions when the vertices of the graph are in general position. We also consider the particular case in which the vertices are in convex position and present bounds for the chromatic index of a few types of decompositions.Comment: 18 pages, 5 figure

Locally constrained homomorphisms on graphs of bounded treewidth and bounded degree.

A homomorphism from a graph G to a graph H is locally bijective, surjective, or injective if its restriction to the neighborhood of every vertex of G is bijective, surjective, or injective, respectively. We prove that the problems of testing whether a given graph G allows a homomorphism to a given graph H that is locally bijective, surjective, or injective, respectively, are NP-complete, even when G has pathwidth at most 5, 4 or 2, respectively, or when both G and H have maximum degree 3. We complement these hardness results by showing that the three problems are polynomial-time solvable if G has bounded treewidth and in addition G or H has bounded maximum degree

Product of digraphs, (super) edge-magic valences and related problems

Discrete Mathematics, and in particular Graph Theory, has gained a lot of popularity during the last 7 decades. Among the many branches in Graph Theory, graph labelings has experimented a fast development, in particular during the last decade. One of the very important type of labelings are super edge-magic labelings introduced in 1998 by Enomoto et al. as a particular case of edge-magic labelings, introduced in 1970 by Kotzig and Rosa. An edge-magic labeling is a bijective mapping from the set of vertices and edges to [1, |V(G)|+|E(G)|], such that the sum of the labels of each edge and the incident vertices to it is constant. The constant is called the valence of the labeling. The edge-magic labeling is called super edge-magic if the smallest labels are assigned to the vertices. In this thesis, we consider three problems related to (super) edge-magic labelings and (di)graph products in which we use a family of super edge-magic digraphs as a second factor of the product. The digraph product we use, the h-product, was introduced by Figueroa-Centeno et al. in 2008. It is a generalization of the Kronecker product of digraphs. In Chapter 2, we study the super edge-magicness of graphs of equal order and size either by providing super edge-magic labelings of some elements in the family or proving that these labelings do not exist. The negative results are specially interesting since these kind of results are not common in the literature. Furthermore, the few results found in this direction usually meet one of the following reasons: too many vertices compared with the number of edges; too many edges compared with the number of vertices; or parity conditions. In our case, all previous reasons fail. In Chapter 3, we enlarge the family of perfect (super) edge-magic crowns. A crown is obtained from a cycle by adding the same number of pendant edges to each vertex of the cycle. Intuitively speaking, a (super) edge-magic graphs is perfect (super) edge-magic if all possible theoretical valences occur. The main result of the chapter is that the crowns defined by a cycle of length pq, where p and q are different odd primes, are perfect (super) edge-magic. We also provided lower bounds for the number of edge-magic valences of crowns. For graphs of equal order and size, the odd and the even labelling construction allows to obtain two edge-magic labelings from a particular super edge-magic labeling. The name refers to the parity of the vertex labels. In Chapter 4, we begin by providing some properties of odd and even labelling construction related to the (super) edge-magic labeling and also with respect to the digraph product. We also get a new application of the h-product by interchanging the role of the factors. This allows us to consider the classical conjecture of Godbold and Slater with respect to valences of cycles with a different point of view than the ones existing. Finally, we devote Chapter 5 to study the problem of edge-magic valences of crowns, in which even cycles appear, and to establish a relationship between super edge-magic graphs and graph decompositions. Some lower bounds on the number of (super) edge-magic valences are also established.La Matemàtica Discreta, i en particular la Teoria de Grafs, han guanyat molta popularitat durant les últimes set dècades. Entre les moltes branques de la Teoria de Grafs, els etiquetatges de grafs han experimentat un ràpid desenvolupament, especialment durant l'última dècada. Un dels tipus d'etiquetatges més importants són els etiquetatges super branca-màgics introduïts el 1998 per Enomoto et al. com un cas particular d'etiquetatges branca-màgics, introduïts el 1970 per Kotzig i Rosa. Un etiquetatge branca-màgic és una aplicació bijectiva del conjunt de vèrtexs i branques a [1, |V(G)|+|E(G)|], de manera que la suma de les etiquetes de cada branca i els vèrtexs incidents a ella és constant. La constant s'anomena valència de l'etiquetatge. L'etiquetatge branca-màgic s'anomena super branca-màgic si les etiquetes més petites s'assignen als vèrtexs. En aquesta tesi, considerem tres problemes relacionats amb etiquetatges (super) branca-màgic i productes de digrafs, en els que intervé una família de grafs super branca-màgic com a segon factor del producte. El producte de digrafs que usem, el producte h, va ser introduït per Figueroa-Centeno et al. el 2008. És una generalització del producte de Kronecker de digraphs. En el Capítol 2, estudiem el caràcter super branca-màgic de grafs d’ordre igual a mida, ja sigui proporcionant etiquetatges super branca-màgics d'alguns elements de la família o demostrant que aquests tipus d’etiquetatges no existeixen. Els resultats negatius són especialment interessants ja que aquest tipus de resultats no són comuns en la literatura. A més, els pocs resultats trobats en aquesta direcció solen encabir-se en una de les raons següents: massa vèrtexs en comparació amb el nombre de branques; massa branques en comparació amb el nombre de vèrtexs; o condicions de paritat. En el nostre cas, totes les raons anteriors fracassen. En el Capítol 3, ampliem la família de corones (super) branca-màgiques perfectes. Una corona és el graf que s’obté a partir d’un afegint el mateix nombre de branques a cada vèrtex del cicle. Intuïtivament parlant, un graf (super) branca màgic és (super) branca màgic si es donen totes les possibles valències teòriques. El resultat principal del capítol és que les corones definides per un cicle de longitud pq, on p i q són primers senars diferents, són (super) branca màgics perfectes. També proporcionem cotes inferiors per a la quantitat de valències màgiques de corones. Per a grafs d'igual ordre i mida, la construcció de l'etiquetatge senar i parell permet obtenir dos etiquetatges branca-màgics a partir d'un etiquetatge super branca-màgic. El nom fa referència a la paritat de les etiquetes de vèrtex. Al capítol 4, comencem proporcionant algunes propietats de la construcció de l'etiquetatge senar i parell relacionades amb l'etiquetatge (super) branca-màgic del que proven i també al producte h de dígrafs. També obtenim una nova aplicació del producte h intercanviant el paper dels factors. Això ens permet considerar la conjectura de Godbold i Slater respecte a les valències dels cicles des d’un punt de vista diferent a les existents. Finalment, dediquem el Capítol 5 a estudiar el problema de les valències branca-màgiques de les corones, en les que apareixen cicles parells, i a establir una relació entre els grafs super branca-màgic i les descomposicions de grafs. També s'estableixen alguns cotes inferiors del nombre de valències (super) branca-màgiques.Postprint (published version

A Linear Time Parameterized Algorithm for Node Unique Label Cover

The optimization version of the Unique Label Cover problem is at the heart of the Unique Games Conjecture which has played an important role in the proof of several tight inapproximability results. In recent years, this problem has been also studied extensively from the point of view of parameterized complexity. Cygan et al. [FOCS 2012] proved that this problem is fixed-parameter tractable (FPT) and Wahlstr\"om [SODA 2014] gave an FPT algorithm with an improved parameter dependence. Subsequently, Iwata, Wahlstr\"om and Yoshida [2014] proved that the edge version of Unique Label Cover can be solved in linear FPT-time. That is, there is an FPT algorithm whose dependence on the input-size is linear. However, such an algorithm for the node version of the problem was left as an open problem. In this paper, we resolve this question by presenting the first linear-time FPT algorithm for Node Unique Label Cover

Sparsest Cut on Bounded Treewidth Graphs: Algorithms and Hardness Results

We give a 2-approximation algorithm for Non-Uniform Sparsest Cut that runs in time $n^{O(k)}$, where $k$ is the treewidth of the graph. This improves on the previous $2^{2^k}$-approximation in time \poly(n) 2^{O(k)} due to Chlamt\'a\v{c} et al. To complement this algorithm, we show the following hardness results: If the Non-Uniform Sparsest Cut problem has a $\rho$-approximation for series-parallel graphs (where $\rho \geq 1$), then the Max Cut problem has an algorithm with approximation factor arbitrarily close to $1/\rho$. Hence, even for such restricted graphs (which have treewidth 2), the Sparsest Cut problem is NP-hard to approximate better than $17/16 - \epsilon$ for $\epsilon > 0$; assuming the Unique Games Conjecture the hardness becomes $1/\alpha_{GW} - \epsilon$. For graphs with large (but constant) treewidth, we show a hardness result of $2 - \epsilon$ assuming the Unique Games Conjecture. Our algorithm rounds a linear program based on (a subset of) the Sherali-Adams lift of the standard Sparsest Cut LP. We show that even for treewidth-2 graphs, the LP has an integrality gap close to 2 even after polynomially many rounds of Sherali-Adams. Hence our approach cannot be improved even on such restricted graphs without using a stronger relaxation

On 1-factorizations of Bipartite Kneser Graphs

It is a challenging open problem to construct an explicit 1-factorization of the bipartite Kneser graph $H(v,t)$, which contains as vertices all $t$-element and $(v-t)$-element subsets of $[v]:=\{1,\ldots,v\}$ and an edge between any two vertices when one is a subset of the other. In this paper, we propose a new framework for designing such 1-factorizations, by which we solve a nontrivial case where $t=2$ and $v$ is an odd prime power. We also revisit two classic constructions for the case $v=2t+1$ --- the \emph{lexical factorization} and \emph{modular factorization}. We provide their simplified definitions and study their inner structures. As a result, an optimal algorithm is designed for computing the lexical factorizations. (An analogous algorithm for the modular factorization is trivial.)Comment: We design the first explicit 1-factorization of H(2,q), where q is a odd prime powe