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The following figures show a graph (left, with blue vertices) and its line graph (right, with green vertices). Each vertex of the line graph is shown labeled with the pair of endpoints of the corresponding edge in the original graph. For instance, the green vertex on the right labeled 1,3 corresponds to the edge on the left between the blue ...
Although this game can be considered as a particular case of the vertex coloring game on line graphs, it is mainly considered in the scientific literature as a distinct game. The game chromatic index of a graph G {\displaystyle G} , denoted by χ g ′ ( G ) {\displaystyle \chi '_{g}(G)} , is the minimum number of colors needed for Alice to win ...
A graph has a k-coloring if and only if it has an acyclic orientation for which the longest path has length at most k; this is the Gallai–Hasse–Roy–Vitaver theorem (Nešetřil & Ossona de Mendez 2012). For planar graphs, vertex colorings are essentially dual to nowhere-zero flows. About infinite graphs, much less is known.
By applying exact algorithms for vertex coloring to the line graph of the input graph, it is possible to optimally edge-color any graph with m edges, regardless of the number of colors needed, in time 2 m m O(1) and exponential space, or in time O(2.2461 m) and only polynomial space (Björklund, Husfeldt & Koivisto 2009).
In the study of graph coloring problems in mathematics and computer science, a greedy coloring or sequential coloring [1] is a coloring of the vertices of a graph formed by a greedy algorithm that considers the vertices of the graph in sequence and assigns each vertex its first available color. Greedy colorings can be found in linear time, but ...
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.
Line chart showing the population of the town of Pushkin, Saint Petersburg from 1800 to 2010, measured at various intervals. A line chart or line graph, also known as curve chart, [1] is a type of chart that displays information as a series of data points called 'markers' connected by straight line segments. [2]
Merging the red and blue colors, and the green and yellow colors, produces a subcoloring with only two colors. In graph theory, a subcoloring is an assignment of colors to a graph's vertices such that each color class induces a vertex disjoint union of cliques. That is, each color class should form a cluster graph.