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Mycielskian construction applied to a 5-cycle graph, producing the Grötzsch graph with 11 vertices and 20 edges, the smallest triangle-free 4-chromatic graph (Chvátal 1974). Let the n vertices of the given graph G be v 1, v 2, . . . , v n. The Mycielski graph μ(G) contains G itself as a subgraph, together with n+1 additional vertices: a ...
The Grötzsch graph is a triangle-free graph that cannot be colored with fewer than four colors. Much research about triangle-free graphs has focused on graph coloring. Every bipartite graph (that is, every 2-colorable graph) is triangle-free, and Grötzsch's theorem states that every triangle-free planar graph may be 3-colored. [8]
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.
For a graph G, let χ(G) denote the chromatic number and Δ(G) the maximum degree of G.The list coloring number ch(G) satisfies the following properties.. ch(G) ≥ χ(G).A k-list-colorable graph must in particular have a list coloring when every vertex is assigned the same list of k colors, which corresponds to a usual k-coloring.
A graceful labeling. Vertex labels are in black, edge labels in red.. In graph theory, a graceful labeling of a graph with m edges is a labeling of its vertices with some subset of the integers from 0 to m 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 ...
Zaker (2006) defines a sequence of graphs called t-atoms, with the property that a graph has Grundy number at least t if and only if it contains a t-atom.Each t-atom is formed from an independent set and a (t − 1)-atom, by adding one edge from each vertex of the (t − 1)-atom to a vertex of the independent set, in such a way that each member of the independent set has at least one edge ...
In an undirected simple graph of order n, the maximum degree of each vertex is n − 1 and the maximum size of the graph is n(n − 1) / 2 . The edges of an undirected simple graph permitting loops G {\displaystyle G} induce a symmetric homogeneous relation ∼ {\displaystyle \sim } on the vertices of G {\displaystyle G} that is called ...
Naserasr showed that every triangle-free planar graph also has a homomorphism to the Clebsch graph, a 4-chromatic graph. By combining these two results, it may be shown that every triangle-free planar graph has a homomorphism to a triangle-free 3-colorable graph, the tensor product of K 3 {\displaystyle K_{3}} with the Clebsch graph.