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Adjacent-vertex-distinguishing-total Coloring
In graph theory, a total coloring is a coloring on the vertices and edges of a graph such that: (1). no adjacent vertices have the same color; (2). no adjacent edges have the same color; and (3). no edge and its endvertices are assigned the same color. In 2005, Zhang et al. added a restriction to the definition of total coloring and proposed a new type of coloring defined as follows. Let ''G'' = (''V'',''E'') be a simple graph endowed with a total coloring φ, and let ''u'' be a vertex of ''G''. The set of colors that occurs in the vertex ''u'' is defined as ''C''(''u'') = ∪ . Two vertices ''u'',''v'' ∈ ''V''(''G'') are distinguishable if their color-sets are distinct, i.e., ''C''(''u'') ≠ ''C''(''v''). In graph theory, a total coloring is an adjacent-vertex-distinguishing-total-coloring (AVD-total-coloring) if it has the following additional property: (4). for every two adjacent vertices ''u'',''v'' of a graph ''G'', their colors-sets are distinct from each other, i.e ... [...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] |
Total Coloring
In graph theory, total coloring is a type of graph coloring on the vertices and edges of a graph. When used without any qualification, a total coloring is always assumed to be ''proper'' in the sense that no adjacent edges, no adjacent vertices and no edge and either endvertex are assigned the same color. The total chromatic number χ″(''G'') of a graph ''G'' is the fewest colors needed in any total coloring of ''G''. 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''). A total coloring is a partitioning of the vertices and edges of the graph into total independent sets. Some inequalities for χ″(''G''): # χ″(''G'') ≥ Δ(''G'') + 1. # χ″(''G'') ≤ Δ(''G'') + 102 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Graph
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction). Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete graphs, with their vertices placed on the points of a regular polygon, had already appeared in the 13th century, in the work of Ramon Llull. Such a drawing is sometimes referred to as a mystic rose. Properties The complete graph on vertices is denoted by . Some sources claim that the letter in this notation stands for the German word , but the German name for a complete graph, , does not contain the letter , and other sources state that the notation honors the contributions of Kazimierz Kuratowski to graph theory. has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. The two sets U and V may be thought of as a coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a triangle: after one node is colored blue and another red, the third vertex of the triangle is connected to vertices of both colors, preventing it from being assigned either color. One often writes G=(U,V,E) to denote a bipartite graph whose partition has the parts U and V, with E denotin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
