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About infinite graphs, much less is known. The following are two of the few results about infinite graph coloring: If all finite subgraphs of an infinite graph G are k-colorable, then so is G, under the assumption of the axiom of choice. This is the de Bruijn–Erdős theorem of de Bruijn & Erdős (1951).
A uniquely total colorable graph is a k-total-chromatic graph that has only one possible (proper) k-total-coloring up to permutation of the colors. Empty graphs, paths, and cycles of length divisible by 3 are uniquely total colorable graphs. Mahmoodian & Shokrollahi (1995) conjectured that they are also the only members in this family.
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.
For any fixed graph K, one considers graphs G that admit a homomorphism to K, written G → K. These are also called K-colorable graphs. This generalizes the usual notion of graph coloring, since it follows from definitions that a k-coloring is the same as a K k-coloring (a homomorphism into the complete graph on k vertices). A graph K is ...
A (k, d)-coloring of a graph G is a coloring of its vertices with k colours such that each vertex v has at most d neighbours having the same colour as the vertex v.We consider k to be a positive integer (it is inconsequential to consider the case when k = 0) and d to be a non-negative integer.
In graph theory, a part of mathematics, a k-partite graph is a graph whose vertices are (or can be) partitioned into k different independent sets. Equivalently, it is a graph that can be colored with k colors, so that no two endpoints of an edge have the same color. When k = 2 these are the bipartite graphs, and when k = 3 they are called the ...
A maximal outerplanar graph and its 3-coloring The complete graph K 4 is the smallest planar graph that is not outerplanar.. In graph theory, an outerplanar graph is a graph that has a planar drawing for which all vertices belong to the outer face of the drawing.
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 ...