Pelabelan tak teratur titik pada graf hasil kali corona antara prisma dan komplemen graf lengkap

INDONESIA: Misalkan G = (V (G), E(G)) adalah suatu graf dan k merupakan suatu bilangan bulat positif. Suatu pelabelan-k total dari G adalah suatu fungsi f∶ V (G) ∪ E(G) → {1,2,...,k}. Bobot titik v berdasarkan pelabelan f dinotasikan dengan w_f (v) dan didefinisikan dengan w_f (v) = f(v) + ∑_(uv∈E(G))▒〖 f(uv)〗. Suatu pelabelan-k total dari G disebut sebagai pelabelan total tak teratur titik ketika tidak ada dua titik berbeda dengan bobot yang sama. Nilai total ketakteraturan titik dari G, dilambangkan dengan tvs(G), adalah nilai k terkecil sehingga G memuat pelabelan-k total tak teratur titik. Misalkan G dan H merupakan suatu graf berorde n dan m, berturut-turut. Hasil kali Corona G⊙H adalah suatu graf yang dibentuk dengan mengambil satu salinan dari G dan n salinan dari H dengan menghubungkan setiap titik pada salinan ke-i dari H ke titik ke-i dari G untuk 1≤i≤n. Pada karya tulis ini, ditentukan nilai total ketakteraturan titik dari graf hasil kali Corona antara graf prisma dengan komplemen graf lengkap. ENGLISH: For a graph G = (V (G), E(G)) and a positive integer k, a total k-labeling of G is a function from V (G) ∪ E(G) to {1,2,...,k}. The vertex weight v by the labeling f defined by wf(v) = f(v) + ∑_(uv∈E(G))▒〖 f(uv)〗. A vertex irregular total k-labeling f is a total k-labeling such that there are no two vertices with the same weight. The total vertex irregularity strength of G is the minimum k such that G has a vertex irregular total k-labeling and denoted by tvs(G), Let G and H be a graph with order n and m, respectively. The corona product G⊙H is a graph obtained by taking one copy of G and n copies of H and join each vertex of the i-th copy of H to the i-th vertex of G, for every 1≤i≤n. In this paper, we determine the exact value of total vertex irregularity strength of Corona product of prism and complement of complete graph

Rotulações graciosas e rotulações semifortes em grafos

Orientador: Christiane Neme CamposTese (doutorado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: Três problemas de rotulação em grafos são investigados nesta tese: a Conjetura das Árvores Graciosas, a Conjetura 1,2,3 e a Conjetura 1,2. Uma rotulação graciosa de um grafo simples G=(V(G),E(G)) é uma função injetora f de V(G) em {0,...,|E(G)|} tal que {|f(u)-f(v)|: uv em E(G)} = {1,...,|E(G)|}. A Conjetura das Árvores Graciosas, proposta por Rosa e Kotzig em 1967, afirma que toda árvore possui uma rotulação graciosa. Um problema relacionado à Conjetura das Árvores Graciosas consiste em determinar se, para todo vértice v de uma árvore T, existe uma rotulação graciosa de T que atribui o rótulo 0 a v. Árvores com tal propriedade são denominadas 0-rotativas. Nesta tese, apresentamos famílias infinitas de caterpillars 0-rotativos. Nossos resultados reforçam a conjetura de que todo caterpillar com diâmetro pelo menos cinco é 0-rotativo. Também investigamos uma rotulação graciosa mais restrita, chamada rotulação-alpha. Uma rotulação graciosa f de G é uma rotulação-alpha se existir um inteiro k, 0 <= k <= |E(G)|, tal que, para toda aresta uv em E(G), f(u) <= k < f(v) ou f(v) <= k < f(u). Nesta tese, apresentamos duas famílias de lobsters com grau máximo três que possuem rotulações-alpha. Nossos resultados contribuem para uma caracterização de todos os lobsters com grau máximo três que possuem rotulações-alpha. Na segunda parte desta tese, investigamos generalizações da Conjetura 1,2,3 e da Conjetura 1,2. Dado um grafo simples G = (V(G),E(G)) e um subconjunto L dos números reais, dizemos que uma função f de E(G) em L é uma L-rotulação de arestas de G e dizemos que uma função f da união de V(G) com E(G) em L é uma L-rotulação total de G. Para todo vértice v de G, a cor de v, C(v), é definida como a soma dos rótulos das arestas incidentes em v, se f for uma L-rotulação de arestas de G. Se f for uma L-rotulação total, C(v) é a soma dos rótulos das arestas incidentes no vértice v mais o valor f(v). O par (f,C) é uma L-rotulação de arestas semiforte (L-rotulação total semiforte) se f for uma rotulação de arestas (rotulação total) e C(u) for diferente de C(v) para quaisquer dois vértices adjacentes u,v de G. A Conjetura 1,2,3, proposta por Karónski et al. em 2004, afirma que todo grafo simples e conexo com pelo menos três vértices possui uma {1,2,3}-rotulação de arestas semiforte. A Conjetura 1,2, proposta por Przybylo e Wozniak em 2010, afirma que todo grafo simples possui uma {1,2}-rotulação total semiforte. Sejam a,b,c três reais distintos. Nesta tese, nós investigamos {a,b,c}-rotulações de arestas semifortes e {a,b}-rotulações totais semifortes para cinco famílias de grafos: as potências de caminho, as potências de ciclo, os grafos split, os grafos cobipartidos regulares e os grafos multipartidos completos. Provamos que essas famílias possuem tais rotulações para alguns valores reais a,b,c. Como corolário de nossos resultados, obtemos que a Conjetura 1,2,3 e a Conjetura 1,2 são verdadeiras para essas famílias. Além disso, também mostramos que nossos resultados em rotulações de arestas semifortes implicam resultados similares para outro problema de rotulação de arestas relacionadoAbstract: This thesis addresses three labelling problems on graphs: the Graceful Tree Conjecture, the 1,2,3-Conjecture, and the 1,2-Conjecture. A graceful labelling of a simple graph G=(V(G),E(G)) is an injective function f from V(G) to {0,...,|E(G)|} such that {|f(u)-f(v)| : uv in E(G)} = {1,...,|E(G)|}. The Graceful Tree Conjecture, posed by Rosa and Kotzig in 1967, states that every tree has a graceful labelling. A problem connected with the Graceful Tree Conjecture consists of determining whether, for every vertex v of a tree T, there exists a graceful labelling of T that assigns label 0 to v. Trees with such a property are called 0-rotatable. In this thesis, we present infinite families of 0-rotatable caterpillars. Our results reinforce a conjecture that states that every caterpillar with diameter at least five is 0-rotatable. We also investigate a stronger type of graceful labelling, called alpha-labelling. A graceful labelling f of G is an alpha-labelling if there exists an integer k with 0<= k <= |E(G)| such that, for each edge uv in E(G), either f(u) <= k < f(v) or f(v) <= k < f(u). In this thesis, we prove that the following families of lobsters have alpha-labellings: lobsters with maximum degree three, without Y-legs and with at most one forbidden ending; and lobsters T with a perfect matching M such that the contracted tree T/M has a balanced bipartition. These results point towards a characterization of all lobsters with maximum degree three that have alpha-labellings. In the second part of the thesis, we focus on generalizations of the 1,2,3-Conjecture and the 1,2-Conjecture. Given a simple graph G=(V(G),E(G)) and a subset L of real numbers, we call a function f from E(G) to L an L-edge-labelling of G, and we call a function f from V(G) union E(G) to L an L-total-labelling of G. For each vertex v of G, the colour of v, C(v), is defined as the sum of the labels of its incident edges, if f is an L-edge-labelling. If f is an L-total-labelling, C(v) is the sum of the labels of the edges incident with vertex v plus the label f(v). The pair (f,C) is a neighbour-distinguishing L-edge-labelling (neighbour-distinguishing L-total-labelling) if f is an edge-labelling (total-labelling) and C(u) is different from C(v), for every edge uv in E(G). The 1,2,3-Conjecture, posed by Kar\'onski et al. in 2004, states that every connected simple graph with at least three vertices has a neighbour-distinguishing {1,2,3}-edge-labelling. The 1,2-Conjecture, posed by Przybylo and Wozniak in 2010, states that every simple graph has a neighbour-distinguishing {1,2}-total-labelling. Let a,b,c be distinct real numbers. In this thesis, we investigate neighbour-distinguishing {a,b,c}-edge-labellings and neighbour-distinguishing {a,b}-total labellings for five families of graphs: powers of paths, powers of cycles, split graphs, regular cobipartite graphs and complete multipartite graphs. We prove that these families have such labellings for some real values a, b, and c. As a corollary of our results, we obtain that the 1,2,3-Conjecture and the 1,2-Conjecture are true for these families. Furthermore, we also show that our results on neighbour-distinguishing edge-labellings imply similar results on a closely related problem called detectable edge-labelling of graphsDoutoradoCiência da ComputaçãoDoutor em Ciência da Computação2014/16861-8FAPESPCAPE

Almost Symmetries and the Unit Commitment Problem

This thesis explores two main topics. The first is almost symmetry detection on graphs. The presence of symmetry in combinatorial optimization problems has long been considered an anathema, but in the past decade considerable progress has been made. Modern integer and constraint programming solvers have automatic symmetry detection built-in to either exploit or avoid symmetric regions of the search space. Automatic symmetry detection generally works by converting the input problem to a graph which is in exact correspondence with the problem formulation. Symmetry can then be detected on this graph using one of the excellent existing algorithms; these are also the symmetries of the problem formulation.The motivation for detecting almost symmetries on graphs is that almost symmetries in an integer program can force the solver to explore nearly symmetric regions of the search space. Because of the known correspondence between integer programming formulations and graphs, this is a first step toward detecting almost symmetries in integer programming formulations. Though we are only able to compute almost symmetries for graphs of modest size, the results indicate that almost symmetry is definitely present in some real-world combinatorial structures, and likely warrants further investigation.The second topic explored in this thesis is integer programming formulations for the unit commitment problem. The unit commitment problem involves scheduling power generators to meet anticipated energy demand while minimizing total system operation cost. Today, practitioners usually formulate and solve unit commitment as a large-scale mixed integer linear program.The original intent of this project was to bring the analysis of almost symmetries to the unit commitment problem. Two power generators are almost symmetric in the unit commitment problem if they have almost identical parameters. Along the way, however, new formulations for power generators were discovered that warranted a thorough investigation of their own. Chapters 4 and 5 are a result of this research.Thus this work makes three contributions to the unit commitment problem: a convex hull description for a power generator accommodating many types of constraints, an improved formulation for time-dependent start-up costs, and an exact symmetry reduction technique via reformulation

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic

Using MapReduce Streaming for Distributed Life Simulation on the Cloud

Distributed software simulations are indispensable in the study of large-scale life models but often require the use of technically complex lower-level distributed computing frameworks, such as MPI. We propose to overcome the complexity challenge by applying the emerging MapReduce (MR) model to distributed life simulations and by running such simulations on the cloud. Technically, we design optimized MR streaming algorithms for discrete and continuous versions of Conway’s life according to a general MR streaming pattern. We chose life because it is simple enough as a testbed for MR’s applicability to a-life simulations and general enough to make our results applicable to various lattice-based a-life models. We implement and empirically evaluate our algorithms’ performance on Amazon’s Elastic MR cloud. Our experiments demonstrate that a single MR optimization technique called strip partitioning can reduce the execution time of continuous life simulations by 64%. To the best of our knowledge, we are the first to propose and evaluate MR streaming algorithms for lattice-based simulations. Our algorithms can serve as prototypes in the development of novel MR simulation algorithms for large-scale lattice-based a-life models.https://digitalcommons.chapman.edu/scs_books/1014/thumbnail.jp

Ahlfors circle maps and total reality: from Riemann to Rohlin

This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known intersection with real algebraic geometry, especially Klein's ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a gallery of pictures quite pleasant to visit of which we have attempted to trace the simplest representatives. This drifted us toward some electrodynamic motions along real circuits of dividing curves perhaps reminiscent of Kepler's planetary motions along ellipses. The ultimate origin of circle maps is of course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass. Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found in Klein (what we failed to assess on printed evidence), the pivotal contribution belongs to Ahlfors 1950 supplying an existence-proof of circle maps, as well as an analysis of an allied function-theoretic extremal problem. Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree controls than available in Ahlfors' era. Accordingly, our partisan belief is that much remains to be clarified regarding the foundation and optimal control of Ahlfors circle maps. The game of sharp estimation may look narrow-minded "Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to contemplate how conformal and algebraic geometry are fighting together for the soul of Riemann surfaces. A second part explores the connection with Hilbert's 16th as envisioned by Rohlin 1978.Comment: 675 pages, 199 figures; extended version of the former text (v.1) by including now Rohlin's theory (v.2