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Exact coloring of the complete graph K 6. Every n-vertex complete graph K n has an exact coloring with n colors, obtained by giving each vertex a distinct color. Every graph with an n-color exact coloring may be obtained as a detachment of a complete graph, a graph obtained from the complete graph by splitting each vertex into an independent set and reconnecting each edge incident to the ...
Once a new vertex has been coloured, the algorithm determines which of the remaining uncoloured vertices has the highest number of colours in its neighbourhood and colours this vertex next. Brélaz defines this number as the degree of saturation of a given vertex. [1] The contraction of the term "degree of saturation" forms the name of the ...
In graph theory, circular coloring is a kind of coloring that may be viewed as a refinement of the usual graph coloring. The circular chromatic number of a graph G {\displaystyle G} , denoted χ c ( G ) {\displaystyle \chi _{c}(G)} can be given by any of the following definitions, all of which are equivalent (for finite graphs).
The total graph T = T(G) of a graph G is a graph such that (i) the vertex set of T corresponds to the vertices and edges of G and (ii) two vertices are adjacent in T if and only if their corresponding elements are either adjacent or incident in G. Then total coloring of G becomes a (proper) vertex coloring of T(G).
Finding ψ(G) is an optimization problem.The decision problem for complete coloring can be phrased as: . INSTANCE: a graph G = (V, E) and positive integer k QUESTION: does there exist a partition of V into k or more disjoint sets V 1, V 2, …, V k such that each V i is an independent set for G and such that for each pair of distinct sets V i, V j, V i ∪ V j is not an independent set.
A Grundy coloring of a t-atom can be obtained by coloring the independent set first with the smallest-numbered color, and then coloring the remaining (t − 1)-atom with an additional t − 1 colors. For instance, the only 1-atom is a single vertex, and the only 2-atom is a single edge, but there are two possible 3-atoms: a triangle and a four ...
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An edge coloring of is called a -rainbow coloring if for every set of vertices of , there is a rainbow tree in containing the vertices of . The k {\displaystyle k} -rainbow index rx k ( G ) {\displaystyle {\text{rx}}_{k}(G)} of G {\displaystyle G} is the minimum number of colors needed in a k {\displaystyle k} -rainbow coloring of G ...