Graceful labeling of triangular extension of complete bipartite graph

For positive integers m, n, K m,n represents the complete bipartite graph. We name the graph G = K m,n ⊙ K2 as triangular extension of complete bipartite graph K m,n , since there is a triangle hanging from every vertex of K m,n . In this paper we show that G is graceful when m = n = 2ℓ, for any integer ℓ

Alpha Labeling of Amalgamated Cycles

A graceful labeling of a bipartite graph is an \a-labeling if it has the property that the labels assigned to the vertices of one stable set of the graph are smaller than the labels assigned to the vertices of the other stable set. A concatenation of cycles is a connected graph formed by a collection of cycles, where each cycle shares at most either two vertices or two edges with other cycles in the collection. In this work we investigate the existence of \a-labelings for this kind of graphs, exploring the concepts of vertex amalgamation to produce a family of Eulerian graphs, and edge amalgamation to generate a family of outerplanar graphs. In addition, we determine the number of graphs obtained with $k$ copies of the cycle $C_n$, for both types of amalgamations

Discussion on some interesting topics in Graph Theory

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Study of Some Interesting Topics in the Theory of Graphs

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Discrete Mathematics and Symmetry

Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group

The Performance Optimization of ASP Solving Based on Encoding Rewriting and Encoding Selection

Answer set programming (ASP) has long been used for modeling and solving hard search problems. These problems are modeled in ASP as encodings, a collection of rules that declaratively describe the logic of the problem without explicitly listing how to solve it. It is common that the same problem has several different but equivalent encodings in ASP. Experience shows that the performance of these ASP encodings may vary greatly from instance to instance when processed by current state-of-the-art ASP grounder/solver systems. In particular, it is rarely the case that one encoding outperforms all others. Moreover, running an ASP system on one encoding for a specific instance may “take forever,” while running it on another encoding for this instance may yield a solution in a fraction of a second. The selection of a ”good” encoding for each instance is crucial to the performance of ASP solving. In this dissertation, I propose methods to improve the performance of ASP solving that exploit these observations. First, I designed and implemented methods that, given an encoding for a problem, rewrite it in several ways into new different but equivalent encodings. Second, I designed and implemented a system that given a set of input encodings of a problem, a set of problem instances, and an ASP grounder/solver system, automatically generates equivalent encodings and builds for each selected encoding its performance model. The model predicts for any instance the execution time that the grounder/solver system takes to process the instance under the corresponding encoding. These performance models are then used to improve solving efficiency: whenever a new instance arrives, the system selects the encoding predicted to perform the best on the instance and invokes the grounder/solver. The system also supports a scheduled execution and an interleaved execution of encodings, which are complementary to machine learning techniques. Third, I implemented algorithms that generate hard structured instances for several combinatorial problems I selected for our experimental study of the efficacy of the methods I developed. Hard instances can serve as the benchmark for evaluating the hardness of specific problems and contribute as training data to the platform I created to help build encoding selection models. The process can also provide meaningful insights into finding hard instances of other combinatorial problems

Graceful labellings of new families of windmill and snake graphs

A function ƒ is a graceful labelling of a graph G = (V,E) with m edges if ƒ is an injection ƒ : V (G) → {0, 1, 2, . . . ,m} such that each edge uv ∈ E is assigned the label |ƒ(u) − ƒ(v)| ∈ {1, 2, . . . ,m}, and no two edge labels are the same. If a graph G has a graceful labelling, we say that G itself is graceful. A variant is a near graceful labelling, which is similar, except the co-domain of f is {0, 1, 2, . . . ,m + 1} and the set of edge labels are either {1, 2, . . . ,m − 1,m} or {1, 2, . . . ,m − 1,m + 1}. In this thesis, we prove any Dutch windmill with three pendant triangles is (near) graceful, which settles Rosa’s conjecture for a new family of triangular cacti. Further, we introduce graceful and near graceful labellings of several families of windmills. In particular, we use Skolem-type sequences to prove (near) graceful labellings exist for windmills with C₃ and C₄ vanes, and infinite families of 3,5-windmills and 3,6-windmills. Furthermore, we offer a new solution showing that the graph obtained from the union of t 5-cycles with one vertex in common (Ct₅ ) is graceful if and only if t ≡ 0,3 (mod 4) and near graceful when t ≡ 1, 2 (mod 4). Also, we present a new sufficiency condition to obtain a graceful labelling for every kC₄ₙ snake and use this condition to label every such snake for n = 1, 2, . . . , 6. Then, we extend this result to cyclic snakes where the cycles lengths vary. Also, we obtain new results on the (near) graceful labelling of cyclic snakes based on cycles of lengths n = 6, 10, 14, completely solving the case n = 6