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Harmonious Coloring
In graph theory, a harmonious coloring is a (proper) vertex coloring in which every pair of colors appears on ''at most'' one pair of adjacent vertices. It is the opposite of the complete coloring, which instead requires every color pairing to occur ''at least'' once. The harmonious chromatic number of a graph is the minimum number of colors needed for any harmonious coloring of . Every graph has a harmonious coloring, since it suffices to assign every vertex a distinct color; thus . There trivially exist graphs with (where is the chromatic number); one example is any path of , which can be 2-colored but has no harmonious coloring with 2 colors. Some properties of : :\chi_(T_) = \left\lceil\frac\right\rceil, where is the complete -ary tree with 3 levels. (Mitchem 1989) Harmonious coloring was first proposed by Harary and Plantholt (1982). Still very little is known about it. See also * Complete coloring In graph theory, a complete coloring is a vertex coloring in whi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonious Coloring Tree
Harmony, in music, is the use of simultaneous pitches (tones, notes), or chords. Harmony or harmonious may also refer to: Computing *Apache Harmony, a Java programming language Open source implementation *ECMAScript Harmony, codename for the sixth edition of the scripting language *Harmony (operating system), an experimental computer operating system developed at the National Research Council of Canada *Harmony (software), a music visualizer program * Harmony (toolkit), a never-completed Qt-like software widget toolkit *HarmonyOS, also known as Hongmeng OS, a IoT and mobile operating system developed by Huawei *Harmony search, an evolutionary algorithm used in optimization problems *Harmony technology, developed by RealNetworks; see *Logitech Harmony, a series of universal remote controls made by Logitech *Project Harmony (FOSS group), a Canonical initiative about contributor agreements for Open Source software * Toon Boom Harmony, a Toon Boom Animation application Fictional cha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are connected by '' edges'' (also called ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a set of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex Coloring
In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color. Vertex coloring is often used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. For example, an edge coloring of a graph is just a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex coloring of its dual. However, non-vertex coloring problems are often stated and studied as-is. This is p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex (graph Theory)
In discrete mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arcs (ordered pairs of vertices). In a diagram of a graph, a vertex is usually represented by a circle with a label, and an edge is represented by a line or arrow extending from one vertex to another. From the point of view of graph theory, vertices are treated as featureless and indivisible objects, although they may have additional structure depending on the application from which the graph arises; for instance, a semantic network is a graph in which the vertices represent concepts or classes of objects. The two vertices forming an edge are said to be the endpoints of this edge, and the edge is said to be incident to the vertices. A vertex ''w'' is said to be ad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Coloring
In graph theory, a complete coloring is a vertex coloring in which every pair of colors appears on ''at least'' one pair of adjacent vertices. Equivalently, a complete coloring is minimal in the sense that it cannot be transformed into a proper coloring with fewer colors by merging pairs of color classes. The achromatic number of a graph is the maximum number of colors possible in any complete coloring of . A complete coloring is the opposite of a harmonious coloring, which requires every pair of colors to appear on ''at most'' one pair of adjacent vertices. Complexity theory Finding is an optimization problem. The decision problem for complete coloring can be phrased as: :INSTANCE: a graph and positive integer :QUESTION: does there exist a partition of into or more disjoint sets such that each is an independent set for and such that for each pair of distinct sets is not an independent set. Determining the achromatic number is NP-hard; determining if it is greater ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph (discrete Mathematics)
In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a Set (mathematics), set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line''). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics. The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chromatic Number
In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color. Vertex coloring is often used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. For example, an edge coloring of a graph is just a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex coloring of its dual. However, non-vertex coloring problems are often stated and studied as-is. This is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Glossary Of Graph Theory
This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges. Symbols A B C D E F G H I K L M N O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Coloring
In graph theory, a complete coloring is a vertex coloring in which every pair of colors appears on ''at least'' one pair of adjacent vertices. Equivalently, a complete coloring is minimal in the sense that it cannot be transformed into a proper coloring with fewer colors by merging pairs of color classes. The achromatic number of a graph is the maximum number of colors possible in any complete coloring of . A complete coloring is the opposite of a harmonious coloring, which requires every pair of colors to appear on ''at most'' one pair of adjacent vertices. Complexity theory Finding is an optimization problem. The decision problem for complete coloring can be phrased as: :INSTANCE: a graph and positive integer :QUESTION: does there exist a partition of into or more disjoint sets such that each is an independent set for and such that for each pair of distinct sets is not an independent set. Determining the achromatic number is NP-hard; determining if it is greater ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonious Labeling
In the mathematical discipline of graph theory, a graph labelling is the assignment of labels, traditionally represented by integers, to edges and/or vertices of a graph. Formally, given a graph , a vertex labelling is a function of to a set of labels; a graph with such a function defined is called a vertex-labeled graph. Likewise, an edge labelling is a function of to a set of labels. In this case, the graph is called an edge-labeled graph. When the edge labels are members of an ordered set (e.g., the real numbers), it may be called a weighted graph. When used without qualification, the term labeled graph generally refers to a vertex-labeled graph with all labels distinct. Such a graph may equivalently be labeled by the consecutive integers , where is the number of vertices in the graph. For many applications, the edges or vertices are given labels that are meaningful in the associated domain. For example, the edges may be assigned weights representing the "cost" of traver ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
