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In the mathematical field of graph theory, the Chvátal graph is an undirected graph with 12 vertices and 24 edges, discovered by Václav Chvátal in 1970. It is the smallest graph that is triangle-free , 4-regular , and 4-chromatic .
In addition, for each edge v i v j of G, the Mycielski graph includes two edges, u i v j and v i u j. Thus, if G has n vertices and m edges, μ(G) has 2n+1 vertices and 3m+n edges. The only new triangles in μ(G) are of the form v i v j u k, where v i v j v k is a triangle in G. Thus, if G is triangle-free, so is μ(G).
In the mathematical field of graph theory, the triangle graph is a planar undirected graph with 3 vertices and 3 edges, in the form of a triangle. [1] The triangle graph is also known as the cycle graph and the complete graph.
In the mathematical discipline of graph theory, Shannon multigraphs, named after Claude Shannon by Vizing (1965), are a special type of triangle graphs, which are used in the field of edge coloring in particular. A Shannon multigraph is multigraph with 3 vertices for which either of the following conditions holds:
In the mathematical field of graph theory, the Grötzsch graph is a triangle-free graph with 11 vertices, 20 edges, chromatic number 4, and crossing number 5. It is named after German mathematician Herbert Grötzsch, who used it as an example in connection with his 1959 theorem that planar triangle-free graphs are 3-colorable.
That is, G is a complete graph on the set V of vertices, and the function w assigns a nonnegative real weight to every edge of G. According to the triangle inequality, for every three vertices u, v, and x, it should be the case that w(uv) + w(vx) ≥ w(ux). Then the algorithm can be described in pseudocode as follows. [1]
The nested triangles graph with two triangles is the graph of the triangular prism, and the nested triangles graph with three triangles is the graph of the triangular bifrustum. More generally, because the nested triangles graphs are planar and 3-vertex-connected , it follows from Steinitz's theorem that they all can be represented as convex ...
The monochromatic triangle problem takes as input an n-node undirected graph G(V,E) with node set V and edge set E. The output is a Boolean value, true if the edge set E of G can be partitioned into two disjoint sets E1 and E2, such that both of the two subgraphs G1(V,E1) and G2(V,E2) are triangle-free graphs, and false otherwise.