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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.
In graph theory, a proper edge coloring of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blue, and green. Edge colorings are one of several different types of graph coloring.
In graph theory, a distinguishing coloring or distinguishing labeling of a graph is an assignment of colors or labels to the vertices of the graph that destroys all of the nontrivial symmetries of the graph. The coloring does not need to be a proper coloring: adjacent vertices are allowed to be given the same color. For the colored graph, there ...
The adjacent-vertex-distinguishing-total-chromatic number χ at (G) of a graph G is the fewest colors needed in an AVD-total-coloring of G. The following lower bound for the AVD-total chromatic number can be obtained from the definition of AVD-total-coloring: If a simple graph G has two adjacent vertices of maximum degree, then χ at ( G ) ≥ ...
Example of a mixed graph Consider adjacent vertices u , v ∈ V {\displaystyle u,v\in V} . A directed edge , called an arc , is an edge with an orientation and can be denoted as u v → {\displaystyle {\overrightarrow {uv}}} or ( u , v ) {\displaystyle (u,v)} (note that u {\displaystyle u} is the tail and v {\displaystyle v} is the head of the ...
The fractional version of the incidence coloring was first introduced by Yang in 2007. An r-tuple incidence k-coloring of a graph G is the assignment of r colors to each incidence of graph G from a set of k colors such that the adjacent incidences are given disjoint sets of colors. [14]
A more general version of the theorem applies to list coloring: given any connected undirected graph with maximum degree Δ that is neither a clique nor an odd cycle, and a list of Δ colors for each vertex, it is possible to choose a color for each vertex from its list so that no two adjacent vertices have the same color. In other words, the ...
The problem is fixed-parameter tractable, meaning that there is an algorithm whose running time can be bounded by a polynomial function of the size of the graph multiplied by a larger function of k. [32] The name odd cycle transversal comes from the fact that a graph is bipartite if and only if it has no odd cycles.