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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 should not exist any one-to-one mapping of the vertices to themselves that preserves both adjacency and coloring. The minimum number of colors in a distinguishing coloring is called the distinguishing number of the graph. Distinguishing colorings and distinguishing numbers were introduced by , who provided the following motivating example, based on a puzzle previously formulated by Frank Rubin: "Suppose you have a ring of keys to different doors; each key only opens one door, but they all look indistinguishable to you. How few colors do you need, in order to color the handles of the keys in such a way that you can uniquely identify each key?"〔. Solution in vol. 12, 1980. As cited by . Instead of using colors, Rubin phrased the problem in terms of key handles that could be distinguished from each other by touch. More precisely, this problem assumes that each key is symmetric, so that the keys cannot be distinguished from each other by their orientations on the keyring.〕 This example is solved by using a distinguishing coloring for a cycle graph. With such a coloring, each key will be uniquely identified by its color and the sequence of colors surrounding it.〔.〕 ==Examples== A graph has distinguishing number one if and only if it is asymmetric.〔See, e.g., 〕 For instance, the Frucht graph has a distinguishing coloring with only one color. In a complete graph, the only distinguishing colorings assign a different color to each vertex. For, if two vertices were assigned the same color, there would exist a symmetry that swapped those two vertices, leaving the rest in place. Therefore, the distinguishing number of the complete graph is . However, the graph obtained from by attaching a degree-one vertex to each vertex of has a significantly smaller distinguishing number, despite having the same symmetry group: it has a distinguishing coloring with colors, obtained by using a different ordered pair of colors for each pair of a vertex and its attached neighbor.〔 For a cycle graph of three, four, or five vertices, three colors are needed to construct a distinguishing coloring. For instance, every two-coloring of a five-cycle has a reflection symmetry. In each of these cycles, assigning a unique color to each of two adjacent vertices and using the third color for all remaining vertices results in a three-color distinguishing coloring. However, cycles of six or more vertices have distinguishing colorings with only two colors. That is, Frank Rubin's keyring puzzle requires three colors for rings of three, four or five keys, but only two colors for six or more keys or for two keys.〔 For instance, in the ring of six keys shown, each key can be distinguished by its color and by the length or lengths of the adjacent blocks of oppositely-colored keys: there is only one key for each combination of key color and adjacent block lengths. Hypercube graphs exhibit a similar phenomenon to cycle graphs. The two- and three-dimensional hypercube graphs (the 4-cycle and the graph of a cube, respectively) have distinguishing number three. However, every hypercube graph of higher dimension has distinguishing number only two.〔.〕 The Petersen graph has distinguishing number 3. However other than this graph and the complete graphs, all Kneser graphs have distinguishing number 2.〔.〕 Similarly, among the generalized Petersen graphs, only the Petersen graph itself and the graph of the cube have distinguishing number 3; the rest have distinguishing number 2.〔. Lal and Bhattacharjya (Theorem 4.3) credit this result to an unpublished masters thesis of K. S. Potanka (Virginia Polytechnic University, 1998).〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Distinguishing coloring」の詳細全文を読む スポンサード リンク
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