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Watkins Snark
In the mathematical field of graph theory, the Watkins snark is a snark with 50 vertices and 75 edges. It was discovered by John J. Watkins in 1989. As a snark, the Watkins graph is a connected, bridgeless cubic graph with chromatic index equal to 4. The Watkins snark is also non-planar and non-hamiltonian. It has book thickness 3 and queue number 2. Another well known snark on 50 vertices is the Szekeres snark, the fifth known snark, discovered by George Szekeres in 1973. Gallery Image:Watkins snark 3COL.svg, The chromatic number of the Watkins snark is 3. Image:Watkins snark 4edge color.svg, The chromatic index of the Watkins snark is 4. Edges 1,2 ,4 ,15 ,3 ,8 ,6 ,37 ,6 ,7 ,10 This list contains selected positive numbers in increasing order, including counts of things, dimensionless quantity, dimensionless quantities and probability, probabilities. Each number is given a name in the Long and short scales, short scale ... ,11 ,22 ,9 ,8 ,12 ,9 ,14 0,1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Watkins Snark
In the mathematical field of graph theory, the Watkins snark is a snark with 50 vertices and 75 edges. It was discovered by John J. Watkins in 1989. As a snark, the Watkins graph is a connected, bridgeless cubic graph with chromatic index equal to 4. The Watkins snark is also non-planar and non-hamiltonian. It has book thickness 3 and queue number 2. Another well known snark on 50 vertices is the Szekeres snark, the fifth known snark, discovered by George Szekeres in 1973. Gallery Image:Watkins snark 3COL.svg, The chromatic number of the Watkins snark is 3. Image:Watkins snark 4edge color.svg, The chromatic index of the Watkins snark is 4. Edges 1,2 ,4 ,15 ,3 ,8 ,6 ,37 ,6 ,7 ,10 This list contains selected positive numbers in increasing order, including counts of things, dimensionless quantity, dimensionless quantities and probability, probabilities. Each number is given a name in the Long and short scales, short scale ... ,11 ,22 ,9 ,8 ,12 ,9 ,14 0,1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chromatic Index
In graph theory, an edge coloring of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blue, and green. Edge colorings are one of several different types of graph coloring. The edge-coloring problem asks whether it is possible to color the edges of a given graph using at most different colors, for a given value of , or with the fewest possible colors. The minimum required number of colors for the edges of a given graph is called the chromatic index of the graph. For example, the edges of the graph in the illustration can be colored by three colors but cannot be colored by two colors, so the graph shown has chromatic index three. By Vizing's theorem, the number of colors needed to edge color a simple graph is either its maximum degree or . For some graphs, such as bipartite graphs and high-degree planar graphs, the number of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1,2], [1,4], [1,15], [2,3], [2,8], [3,6], [3,37], [4,6], [4,7], [5,10], [5,11], [5,22], [6,9], [7,8], [7,12], [8,9], [9,14], [10,13], [10,17], [11,16], [11,18], [12,14], [12,33], [13,15], [13,16], [14,20], [15,21], [16,19], [17,18], [17,19], [18,30], [19,21], [20,24], [20,26], [21,50], [22,23], [22,27], [23,24], [23,25], [24,29], [25,26], [25,28], [26,31], [27,28], [27,48], [28,29], [29,31], [30,32], [30,36], [31,36], [32,34], [32,35], [33,34], [33,40], [34,41], [35,38], [35,40], [36,38], [37,39], [37,42], [38,41], [39,44], [39,46], [40,46], [41,46], [42,43], [42,45], [43,44], [43,49], [44,47], [45,47], [45,48], [47,50], [48,49], [49,50
Onekama ( ) is a village in Manistee County in the U.S. state of Michigan. The population was 411 at the 2010 census. The village is located on the shores of Portage Lake and is surrounded by Onekama Township. The town's name is derived from "Ona-ga-maa," an Anishinaabe word which means "singing water." Geography According to the United States Census Bureau, the village has a total area of , all land. The M-22 highway runs through downtown Onekama. History The predecessor of the village of Onekama was the settlement of Portage at Portage Point, first established in 1845, at the western end of Portage, at the outlet of Portage Creek. In 1871, when landowners around the land-locked lake became exasperated with the practices of the Portage Sawmill, they took the solution into their own hands and dug a channel through the narrow isthmus, opening a waterway that lowered the lake by 12 to 14 feet and brought it to the same level as Lake Michigan. When this action dried out Portage ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chromatic Number
In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color. Vertex coloring is often used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. For example, an edge coloring of a graph is just a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex coloring of its dual. However, non-vertex coloring problems are often stated and studied as-is. This is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George Szekeres
George Szekeres AM FAA (; 29 May 1911 – 28 August 2005) was a Hungarian–Australian mathematician. Early years Szekeres was born in Budapest, Hungary, as Szekeres György and received his degree in chemistry at the Technical University of Budapest. He worked six years in Budapest as an analytical chemist. He married Esther Klein in 1937.Obituary The Sydney Morning Herald Being , the family had to escape from the persecution so Szekeres took a job in Shanghai, China. There they lived through World War II, the Japanese occupation and the beginn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Szekeres Snark
In the mathematical field of graph theory, the Szekeres snark is a snark with 50 vertices and 75 edges. It was the fifth known snark, discovered by George Szekeres in 1973. As a snark, the Szekeres graph is a connected, bridgeless cubic graph with chromatic index equal to 4. The Szekeres snark is non-planar and non-hamiltonian but is hypohamiltonian. It has book thickness 3 and queue number 2. Another well known snark on 50 vertices is the Watkins snark discovered by John J. Watkins in 1989.Watkins, J. J. "Snarks." Ann. New York Acad. Sci. 576, 606-622, 1989. Gallery Image:Szekeres snark 3COL.svg, The chromatic number of the Szekeres snark is 3. Image:Szekeres snark 4color edge.svg, The chromatic index In graph theory, an edge coloring of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blue ... of the Szekeres snark ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian Graph
In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex exactly once. A Hamiltonian path that starts and ends at adjacent vertices can be completed by adding one more edge to form a Hamiltonian cycle, and removing any edge from a Hamiltonian cycle produces a Hamiltonian path. Determining whether such paths and cycles exist in graphs (the Hamiltonian path problem and Hamiltonian cycle problem) are NP-complete. Hamiltonian paths and cycles are named after William Rowan Hamilton who invented the icosian game, now also known as ''Hamilton's puzzle'', which involves finding a Hamiltonian cycle in the edge graph of the dodecahedron. Hamilton solved this problem using the icosian calculus, an algebraic structure based on roots of unity with many similarities to the quaternions (also invented by Hami ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planar Graph
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points. Every graph that can be drawn on a plane can be drawn on the sphere as well, and vice versa, by means of stereographic projection. Plane graphs can be encoded by combinatorial maps or rotation systems. An equivalence class of topologically equivalent drawings on the sphere, usually with additional assumptions such as the absence of isthmuses, is called a pl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Graph
In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are also called trivalent graphs. A bicubic graph is a cubic bipartite graph. Symmetry In 1932, Ronald M. Foster began collecting examples of cubic symmetric graphs, forming the start of the Foster census.. Many well-known individual graphs are cubic and symmetric, including the utility graph, the Petersen graph, the Heawood graph, the Möbius–Kantor graph, the Pappus graph, the Desargues graph, the Nauru graph, the Coxeter graph, the Tutte–Coxeter graph, the Dyck graph, the Foster graph and the Biggs–Smith graph. W. T. Tutte classified the symmetric cubic graphs by the smallest integer number ''s'' such that each two oriented paths of length ''s'' can be mapped to each other by exactly one symmetry of the graph. He showed that ''s'' is at most 5, and provided examples of graphs with each possible ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Snark (graph Theory)
In the mathematical field of graph theory, a snark is an undirected graph with exactly three edges per vertex whose edges cannot be colored with only three colors. In order to avoid trivial cases, snarks are often restricted to have additional requirements on their connectivity and on the length of their cycles. Infinitely many snarks exist. One of the equivalent forms of the four color theorem is that every snark is a non-planar graph. Research on snarks originated in Peter G. Tait's work on the four color theorem in 1880, but their name is much newer, given to them by Martin Gardner in 1976. Beyond coloring, snarks also have connections to other hard problems in graph theory: writing in the ''Electronic Journal of Combinatorics'', Miroslav Chladný and Martin Škoviera state that As well as the problems they mention, W. T. Tutte's ''snark conjecture'' concerns the existence of Petersen graphs as graph minors of snarks; its proof has been long announced but remains unp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bridge (graph Theory)
In graph theory, a bridge, isthmus, cut-edge, or cut arc is an edge of a graph whose deletion increases the graph's number of connected components. Equivalently, an edge is a bridge if and only if it is not contained in any cycle. For a connected graph, a bridge can uniquely determine a cut. A graph is said to be bridgeless or isthmus-free if it contains no bridges. This type of bridge should be distinguished from an unrelated meaning of "bridge" in graph theory, a subgraph separated from the rest of the graph by a specified subset of vertices; see . Trees and forests A graph with n nodes can contain at most n-1 bridges, since adding additional edges must create a cycle. The graphs with exactly n-1 bridges are exactly the trees, and the graphs in which every edge is a bridge are exactly the forests. In every undirected graph, there is an equivalence relation on the vertices according to which two vertices are related to each other whenever there are two edge-disjoint paths c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ortrud Oellermann
Ortrud R. Oellermann is a South African mathematician specializing in graph theory. She is a professor of mathematics at the University of Winnipeg. Education and career Oellermann was born in Vryheid. She earned a bachelor's degree, ''cum laude'' honours, and a master's degree at the University of Natal in 1981, 1982, and 1983 respectively, as a student of Henda Swart. She completed her Ph.D. in 1986 at Western Michigan University. Her dissertation was ''Generalized Connectivity in Graphs'' and was supervised by Gary Chartrand. Oellermann taught at the University of Durban-Westville, Western Michigan University, University of Natal, and Brandon University, before moving to Winnipeg in 1996. At Winnipeg, she was co-chair of mathematics and statistics for 2011–2013. Contributions With Gary Chartrand, Oellermann is the author of the book ''Applied and Algorithmic Graph Theory'' (McGraw Hill, 1993). She is also the author of well-cited research publications on metric dimension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
