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In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. Adjacent means that two regions share a common boundary of non-zero length (i.e., not merely a corner where three or more regions meet). [1]
Given a graph G and given a set L(v) of colors for each vertex v (called a list), a list coloring is a choice function that maps every vertex v to a color in the list L(v).As with graph coloring, a list coloring is generally assumed to be proper, meaning no two adjacent vertices receive the same color.
An edge coloring of a graph is a proper coloring of the edges, meaning an assignment of colors to edges so that no vertex is incident to two edges of the same color. An edge coloring with k colors is called a k-edge-coloring and is equivalent to the problem of partitioning the edge set into k matchings.
The edge-coloring problem asks whether it is possible to color the edges of a given graph using at most k different colors, for a given value of k, or with the fewest possible colors. The minimum required number of colors for the edges of a given graph is called the chromatic index of the graph. For example, the edges of the graph in the ...
Since the graph is a triangulation, the vertices adjacent to must form a cycle, and since it has only degree 7, the degree 5 neighbors cannot be all separated by vertices of higher degree; at least two of the degree 5 neighbors of must be adjacent to each other on this cycle. This yields the light edge.
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 ) ≥ ...
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] By definition, it is obvious that 1-tuple incidence k-coloring is an incidence k-coloring too.
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 ...