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Triangle-free
In the mathematics, mathematical area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle graph, triangle of edges. Triangle-free graphs may be equivalently defined as graphs with clique (graph theory), clique number ≤ 2, graphs with girth (graph theory), girth ≥ 4, graphs with no induced path, induced 3-cycle, or Neighbourhood (graph theory), locally independent graphs. By Turán's theorem, the ''n''-vertex triangle-free graph with the maximum number of edges is a complete bipartite graph in which the numbers of vertices on each side of the bipartition are as equal as possible. Triangle finding problem The triangle finding or triangle detection problem is the problem of determining whether a graph is triangle-free or not. When the graph does contain a triangle, algorithms are often required to output three vertices which form a triangle in the graph. It is possible to test whether a graph with m edges i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grötzsch's Theorem
In the mathematics, mathematical field of graph theory, Grötzsch's theorem is the statement that every triangle-free graph, triangle-free planar graph can be graph coloring, colored with only three colors. According to the four-color theorem, every graph that can be drawn in the plane without edge crossings can have its vertices colored using at most four different colors, so that the two endpoints of every edge have different colors, but according to Grötzsch's theorem only three colors are needed for planar graphs that do not contain three mutually adjacent vertices. History The theorem is named after German mathematician Herbert Grötzsch, who published its proof in 1959. Grötzsch's original proof was complex. attempted to simplify it but his proof was erroneous.. In 1989, Richard Steinberg and Dan Younger formulated and proved a planar dual version of the theorem: a 3-edge-connected planar graph (or more generally a planar graph with no bridges and at most three 3-edge cut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
