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Claw-free graphs are the graphs that are locally co-triangle-free; that is, for all vertices, the complement graph of the neighbourhood of the vertex does not contain a triangle. 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 ...
A drawing of a graph with 6 vertices and 7 edges.. In mathematics and computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.
In graph theory, Vizing's theorem states that every simple undirected graph may be edge colored using a number of colors that is at most one larger than the maximum degree Δ of the graph. At least Δ colors are always necessary, so the undirected graphs may be partitioned into two classes: "class one" graphs for which Δ colors suffice, and ...
For instance, R 2 (3) = 6, that is, if the edges of the graph K 6 are 2-colored, there will always be a monochromatic triangle. A path in an edge-colored graph is said to be a rainbow path if no color repeats on it. A graph is said to be rainbow colored if there is a rainbow path between any two pairs of vertices.
Whenever two edges of a Euclidean minimum spanning tree meet at a vertex, they must form an angle of 60° or more, with equality only when they form two sides of an equilateral triangle. This is because, for two edges forming any sharper angle, one of the two edges could be replaced by the third, shorter edge of the triangle they form, forming ...
A line perfect graph. The edges in each biconnected component are colored black if the component is bipartite, blue if the component is a tetrahedron, and red if the component is a book of triangles. The line graph of the complete graph K n is also known as the triangular graph, the Johnson graph J(n, 2), or the complement of the Kneser graph ...
This nine-edge Shannon multigraph requires nine colors in any edge coloring; its vertex degree is six and its multiplicity is three. According to a theorem of Shannon (1949) , every multigraph with maximum degree Δ {\displaystyle \Delta } has an edge coloring that uses at most 3 2 Δ {\displaystyle {\frac {3}{2}}\Delta } colors.
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]
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