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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]
The graph of the 3-3 duoprism (the line graph of ,) is perfect. Here it is colored with three colors, with one of its 3-vertex maximum cliques highlighted. In graph theory, a perfect graph is a graph in which the chromatic number equals the size of the maximum clique, both in the graph itself and in every induced subgraph. In all graphs, the ...
A pie chart showing the composition of the 38th Parliament of Canada. A chart (sometimes known as a graph) is a graphical representation for data visualization, in which "the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart". [1]
Pseudoforest, a graph in which each connected component has at most one cycle; Strangulated graph, a graph in which every peripheral cycle is a triangle; Strongly connected graph, a directed graph in which every edge is part of a cycle; Triangle-free graph, a graph without three-vertex cycles; Even-cycle-free graph, a graph without even cycles
A graph that is locally H is claw-free if and only if the independence number of H is at most two; for instance, the graph of the regular icosahedron is claw-free because it is locally C 5 and C 5 has independence number two. The locally linear graphs are the graphs in which every neighbourhood is an induced matching. [5]
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.