Clinton, for her part, has repeatedly declined opportunities to call Trump himself a racist, despite repeated questioning on the subject and multiple labelings of some of his campaign statements as racist by her campaign.
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"When you look at the data, and those data are compelling, they tell us that we do have a crisis at our hands and we have an epidemic, and if we're going to utilize those labelings, let's act accordingly, and let's respond in crisis proportion," Tonko, who sits on the House Energy and Commerce Committee, told The Hill's editor-in-chief Bob CusackRobert (Bob) CusackDemocrat: Lawmakers need to approach opioid crisis as 'a chronic situation' The Hill's Editor-in-Chief: Can Michael Bloomberg erase his terrible debate performance?
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Both versions reduce to Sperner's lemma when m=1, or when all m labelings are identical. See for similar generalizations.
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The friendly index set of , denoted , is the set of numbers that can arise as friendly indexes of friendly labelings of . The Dynamic Survey of Graph Labeling contains a list of papers that examines the friendly indices of various graphs.
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With Walter Wallis, Marr is the author of a book on magic graphs and graph labeling, Magic Graphs (2nd ed., Springer, 2013). She spoke about magic graph labelings as an invited speaker at the Midwest Conference on Combinatorics, Cryptography, and Computing in 2011.
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In graph theory, an edge-graceful graph labeling is a type of graph labeling. This is a labeling for simple graphs in which no two distinct edges connect the same two distinct vertices, no edge connects a vertex to itself, and the graph is connected. Edge-graceful labelings were first introduced by Sheng- Ping Lo in his seminal paper.
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Appending an edge and a vertex to P2 gives P3, the path with three vertices. Denote the vertices by v1, v2, and v3. Label the two edges in the following way: the edge (v1, v2) is labeled 1 and (v2, v3) labeled 2. The induced labelings on v1, v2, and v3 are then 1, 0, and 2 respectively.
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Marr graduated from Murray State University in 2002, and earned a master's degree in mathematics at Texas A&M; University in 2004. She completed her Ph.D. in 2007 at Southern Illinois University; her dissertation, Labelings of Directed Graphs, was supervised by Walter D. Wallis. She has been a member of the mathematics faculty at Southwestern University since 2007. She was department chair for 2015–2018.
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In chemical graph theory and in mathematical chemistry, a molecular graph or chemical graph is a representation of the structural formula of a chemical compound in terms of graph theory. A chemical graph is a labeled graph whose vertices correspond to the atoms of the compound and edges correspond to chemical bonds. Its vertices are labeled with the kinds of the corresponding atoms and edges are labeled with the types of bonds. For particular purposes any of the labelings may be ignored.
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Also, the same numbers are used to represent both pitches and intervals. For example, the number 4 serves both as a label for the pitch class E (if C = 0) and as a label for the distance between the pitch classes D and F. (In much the same way, the term "10 degrees" can label both a temperature and the distance between two temperatures.) Only one of these labelings is sensitive to the (arbitrary) choice of pitch class 0. For example, if one makes a different choice about which pitch class is labeled 0, then the pitch class E will no longer be labeled "4". However, the distance between D and F will still be assigned the number 4.
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Zwetschgenkuchen, Pflaumenkuchen or Zwetschgendatschi (southern Bavaria) is a sheet cake or pie made from yeast dough or shortcrust dough that is thinly spread onto a baking sheet or other baking mold and covered with pitted zwetschgen plums. It is popular as a summer cake and has different local labelings throughout Germany, Austria and Switzerland. Augsburger Zwetschgendatschi In Hessen, Rhineland-Palatinate, Saarland and Moselle it is known as Quetschekuche, in Bavaria, Baden-Württemberg and parts of Austria it is called Zwetschgendatschi and in Rhineland and the Eifel Prummetaat. "Datschi" is thought to be derived from the dialect word "detschen" or "datschen" that can be translated as "pinching" (as the plums are pinched into the dough).
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Zhaneta Byberi, daughter of Petrit Byberi and Aishe Kurmekaj, was born in Bajram Curri, Tropoja, Albania. Zhaneta was forced to leave Albania at the age of 4, due to numerous conflicts and random family labelings as 'dangerous' and 'anti-nationalistic', thus her family members being politically pursued and repressed would lead to the result of being sent to camps risking the life of her parents and her very self with the continuation of living in Albania at that period. It was not until in 1997 that the government collapsed which caused disorder and rebellion throughout the country. The government attempted to suppress the rebellion by military force but the attempt failed, due to long-term corruption of the armed forces.
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The "pearls" of the title include theorems, proofs, problems, and examples in graph theory. It has ten chapters; after an introductory chapter on basic definitions, the remaining chapters material on graph coloring; Hamiltonian cycles and Euler tours; extremal graph theory; subgraph counting problems including connections to permutations, derangements, and Cayley's formula; graph labelings; planar graphs, the four color theorem, and the circle packing theorem; near-planar graphs; and graph embeddings on topological surfaces. The book also includes several unsolved problems such as the Oberwolfach problem on covering complete graphs by cycles, the characterization of magic graphs, and ringel's "earth-moon" problem on coloring biplanar graphs. Despite its subtitle promising "a comprehensive introduction" to graph theory, many important topics in graph theory are not covered, with the selection of topics reflecting author Ringel's research interests.
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The cycle space of a graph may be interpreted using the theory of homology as the homology group H_1(G,\Z_2) of a simplicial complex with a point for each vertex of the graph and a line segment for each edge of the graph. This construction may be generalized to the homology group H_1(G,R) over an arbitrary ring R. An important special case is the ring of integers, for which the homology group H_1(G,\Z) is a free abelian group, a subgroup of the free abelian group generated by the edges of the graph. Less abstractly, this group can be constructed by assigning an arbitrary orientation to the edges of the given graph; then the elements of H_1(G,\Z) are labelings of the edges of the graph by integers with the property that, at each vertex, the sum of the incoming edge labels equals the sum of the outgoing edge labels. The group operation is addition of these vectors of labels.
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Miller was the author of two books on magic graphs, Super Edge-Antimagic Graphs: A Wealth of Problems and Some Solutions (with Martin Bača, BrownWalker Press, 2008), and (posthumously) Magic and Antimagic Graphs: Attributes, Observations and Challenges in Graph Labelings (with Bača, Joe Ryan, and Andrea Semaničová-Feňovčíková, Springer, 2019). She wrote over 200 research publications, including a widely cited survey of the degree diameter problem, supervised 20 doctoral students before her death, was the supervisor of six more at the time of her death, and helped found four workshop series on algorithms, graph theory, and networks. She was also influential in the history of graph theory in Indonesia, where she visited twice and supervised six doctoral students. An infinite family of vertex-transitive graphs with diameter two and a large number of vertices relative to their degree and diameter, the McKay–Miller–Širáň graphs, are named after Miller and her co- authors Brendan McKay and Jozef Širáň, who first constructed them in 1998.
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The name "graceful labeling" is due to Solomon W. Golomb; this type of labeling was originally given the name β-labeling by Alexander Rosa in a 1967 paper on graph labelings.. A major conjecture in graph theory is the Graceful Tree conjecture or Ringel–Kotzig conjecture, named after Gerhard Ringel and Anton Kotzig, which hypothesizes that all trees are graceful. It is still an open conjecture, although a related but slightly weaker conjecture known as Ringel's conjecture was proven true in 2020.. The Ringel–Kotzig conjecture is also known as the "graceful labeling conjecture". Kotzig once called the effort to prove the conjecture a "disease".. Another weaker version of graceful labelling is the near graceful labeling, in which the vertices can be labeled using some subset of the integers between 0 and m+1 inclusive, such that no two vertices share a label, and each edge is uniquely identified by the absolute difference between its endpoints, such that this magnitude lies between 1 and m+1 inclusive. Another conjecture in graph theory is the Rosa's Conjecture, named after Alexander Rosa, which says that all triangular cacti are graceful or nearly-graceful..
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In graph theory, a branch of mathematics, graph canonization is the problem finding a canonical form of a given graph G. A canonical form is a labeled graph Canon(G) that is isomorphic to G, such that every graph that is isomorphic to G has the same canonical form as G. Thus, from a solution to the graph canonization problem, one could also solve the problem of graph isomorphism: to test whether two graphs G and H are isomorphic, compute their canonical forms Canon(G) and Canon(H), and test whether these two canonical forms are identical. The canonical form of a graph is an example of a complete graph invariant: every two isomorphic graphs have the same canonical form, and every two non-isomorphic graphs have different canonical forms... Conversely, every complete invariant of graphs may be used to construct a canonical form.. The vertex set of an n-vertex graph may be identified with the integers from 1 to n, and using such an identification a canonical form of a graph may also be described as a permutation of its vertices. Canonical forms of a graph are also called canonical labelings,. and graph canonization is also sometimes known as graph canonicalization.
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