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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; Even-hole-free graph, a graph without even cycles of length larger or equal to 6
Vertex coloring is often used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. For example, an edge coloring of a graph is just a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex coloring of its dual. However, non-vertex coloring ...
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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.
2. A color class of a colored graph is the set of vertices or edges having one particular color. 3. In the context of Vizing's theorem, on edge coloring simple graphs, a graph is said to be of class one if its chromatic index equals its maximum degree, and class two if its chromatic index equals one plus the degree. According to Vizing's ...
The 7 cycles of the wheel graph W 4. For odd values of n, W n is a perfect graph with chromatic number 3: the vertices of the cycle can be given two colors, and the center vertex given a third color. For even n, W n has chromatic number 4, and (when n ≥ 6) is not perfect. W 7 is the only wheel graph that is a unit distance graph in the ...
A directed cycle graph of length 8. A directed cycle graph is a directed version of a cycle graph, with all the edges being oriented in the same direction. In a directed graph, a set of edges which contains at least one edge (or arc) from each directed cycle is called a feedback arc set.
A distinguishing coloring of a ring of six keys, using two colors (red and unpainted) For a cycle graph of three, four, or five vertices, three colors are needed to construct a distinguishing coloring. For instance, every two-coloring of a five-cycle has a reflection symmetry.