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A graph with edges colored to illustrate a closed walk, H–A–B–A–H, in green; a circuit which is a closed walk in which all edges are distinct, B–D–E–F–D–C–B, in blue; and a cycle which is a closed walk in which all vertices are distinct, H–D–G–H, in red.
An unlabeled coloring of a graph is an orbit of a coloring under the action of the automorphism group of the graph. The colors remain labeled; it is the graph that is unlabeled. There is an analogue of the chromatic polynomial which counts the number of unlabeled colorings of a graph from a given finite color set.
The star chromatic number of G is the fewest colors needed to star color G. One generalization of star coloring is the closely related concept of acyclic coloring , where it is required that every cycle uses at least three colors, so the two-color induced subgraphs are forests .
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.
Rainbow coloring of a wheel graph, with three colors.Every two non-adjacent vertices can be connected by a rainbow path, either directly through the center vertex (bottom left) or by detouring around one triangle to avoid a repeated edge color (bottom right).
This would have established the four color theorem. No graph can be 0-colored, so 0 is always a chromatic root. Only edgeless graphs can be 1-colored, so 1 is a chromatic root of every graph with at least one edge. On the other hand, except for these two points, no graph can have a chromatic root at a real number smaller than or equal to 32/27. [8]
The girth of a graph is the length of its shortest cycle. graph The fundamental object of study in graph theory, a system of vertices connected in pairs by edges. Often subdivided into directed graphs or undirected graphs according to whether the edges have an orientation or not. Mixed graphs include both types of edges. greedy