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Graph Coloring Game
The graph coloring game is a mathematical game related to graph theory. Coloring game problems arose as game-theoretic versions of well-known graph coloring problems. In a coloring game, two players use a given set of colors to construct a coloring of a Graph (discrete mathematics), graph, following specific rules depending on the game we consider. One player tries to successfully complete the coloring of the graph, when the other one tries to prevent him from achieving it. Vertex coloring game The vertex coloring game was introduced in 1981 by Steven Brams as a map-coloring game and rediscovered ten years after by Bodlaender. Its rules are as follows: # Alice and Bob color the vertices of a graph ''G'' with a set ''k'' of colors. # Alice and Bob take turns, graph coloring#Vertex coloring, coloring properly an uncolored vertex (in the standard version, Alice begins). # If a vertex ''v'' is impossible to color properly (for any color, ''v'' has a neighbor colored with it), the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Clique (graph Theory)
In graph theory, a clique ( or ) is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent. That is, a clique of a graph G is an induced subgraph of G that is complete. Cliques are one of the basic concepts of graph theory and are used in many other mathematical problems and constructions on graphs. Cliques have also been studied in computer science: the task of finding whether there is a clique of a given size in a graph (the clique problem) is NP-complete, but despite this hardness result, many algorithms for finding cliques have been studied. Although the study of complete subgraphs goes back at least to the graph-theoretic reformulation of Ramsey theory by , the term ''clique'' comes from , who used complete subgraphs in social networks to model cliques of people; that is, groups of people all of whom know each other. Cliques have many other applications in the sciences and particularly in bioinformatics. Definiti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
(a,b)-decomposability
In graph theory, the (''a'', ''b'')-decomposition of an undirected graph is a partition of its edges into ''a'' + 1 sets, each one of them inducing a forest, except one which induces a graph with maximum degree ''b''. If this graph is also a forest, then we call this a F(''a'', ''b'')-decomposition. A graph with arboricity ''a'' is (''a'', 0)-decomposable. Every (''a'', ''0'')-decomposition or (''a'', ''1'')-decomposition is a F(''a'', ''0'')-decomposition or a F(''a'', ''1'')-decomposition respectively. Graph classes * Every planar graph is F(2, 4)-decomposable. * Every planar graph G with girth at least g is ** F(2, 0)-decomposable if g \ge 4.Implied by . ** (1, 4)-decomposable if g \ge 5. ** F(1, 2)-decomposable if g \ge 6. ** F(1, 1)-decomposable if g \ge 8, or if every cycle of G is either a triangle or a cycle with at least 8 edges not belonging to a triangle. ** (1, 5)-decomposable if G has n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Incidence (graph)
In graph theory, a vertex is incident with an edge if the vertex is one of the two vertices the edge connects. An incidence is a pair (u, e) where u is a vertex and e is an edge incident with u. Two distinct incidences (u, e) and (v,f) are adjacent if and only if u = v, e = f or uv = e or f. An incidence coloring In graph theory, the act of Graph coloring, coloring generally implies the assignment of labels to Vertex (graph theory), vertices, Glossary of graph theory terms#edge, edges or Glossary of graph theory terms#face, faces in a graph (discrete mathem ... of a graph G is an assignment of a color to each incidence of G in such a way that adjacent incidences get distinct colors. It is equivalent to a strong edge coloring of the graph obtained by subdivising each edge of G once. References The Incidence Coloring Pageby Éric Sopena. Graph theory objects {{graph-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Incidence Coloring
In graph theory, the act of Graph coloring, coloring generally implies the assignment of labels to Vertex (graph theory), vertices, Glossary of graph theory terms#edge, edges or Glossary of graph theory terms#face, faces in a graph (discrete mathematics), graph. The incidence coloring is a special graph labeling where each Incidence (graph), incidence of an edge with a vertex is assigned a color under certain constraints. Definitions Below ''G'' denotes a Graph (discrete mathematics), simple graph with non-empty vertex Set (mathematics), set (non-empty) ''V''(''G''), edge set ''E''(''G'') and Degree (graph theory), maximum degree Δ(''G''). Definition. An incidence (graph), incidence is defined as a pair (''v'', ''e'') where v\in V(G) is an end point of e\in E(G). In simple words, one says that vertex ''v'' is incident to edge ''e''. Two incidences (''v'', ''e'') and (''u'', ''f'') are said to be adjacent or neighboring if one of the following holds: * ''v'' = ''u'', ''e'' ≠ ''f' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Caterpillar Tree
In graph theory, a caterpillar or caterpillar tree is a tree (graph theory), tree in which all the Vertex (graph theory), vertices are within distance 1 of a central Path graph, path. Caterpillars were first studied in a series of papers by Harary and Schwenk. The name was suggested by Arthur Hobbs (mathematician), Arthur Hobbs. As colorfully write, "A caterpillar is a tree which metamorphoses into a path when its cocoon of endpoints is removed.". Equivalent characterizations The following characterizations all describe the caterpillar trees: *They are the trees for which removing the leaves and incident edges produces a path graph. *They are the trees in which there exists a path that contains every vertex of degree two or more. *They are the trees in which every vertex of degree at least three has at most two non-leaf neighbors. *They are the trees that do not contain as a subgraph the graph formed by replacing every edge in the star graph ''K''1,3 by a path of length two. *T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Wheel Graph
In graph theory, a wheel graph is a graph formed by connecting a single universal vertex to all vertices of a cycle. A wheel graph with vertices can also be defined as the 1-skeleton of an pyramid. Some authors write to denote a wheel graph with vertices (); other authors instead use to denote a wheel graph with vertices (), which is formed by connecting a single vertex to all vertices of a cycle of length . The former notation is used in the rest of this article and in the table on the right. Edge Set would be the edge set of a wheel graph with vertex set in which the vertex 1 is a universal vertex. Properties Wheel graphs are planar graphs, and have a unique planar embedding. More specifically, every wheel graph is a Halin graph. They are self-dual: the planar dual of any wheel graph is an isomorphic graph. Every maximal planar graph, other than ''K''4 = ''W''4, contains as a subgraph either ''W''5 or ''W''6. There is always a Hamiltonian cycle in the wheel graph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Line Graph
In the mathematics, mathematical discipline of graph theory, the line graph of an undirected graph is another graph that represents the adjacencies between edge (graph theory), edges of . is constructed in the following way: for each edge in , make a vertex in ; for every two edges in that have a vertex in common, make an edge between their corresponding vertices in . The name ''line graph'' comes from a paper by although both and used the construction before this. Other terms used for the line graph include the covering graph, the derivative, the edge-to-vertex dual, the conjugate, the representative graph, and the θ-obrazom, as well as the edge graph, the interchange graph, the adjoint graph, and the derived graph., p. 71. proved that with one exceptional case the structure of a connected graph can be recovered completely from its line graph. Many other properties of line graphs follow by translating the properties of the underlying graph from vertices into edges ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Edge Coloring
In graph theory, a proper edge coloring of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blue, and green. Edge colorings are one of several different types of graph coloring. The edge-coloring problem asks whether it is possible to color the edges of a given graph using at most different colors, for a given value of , 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 illustration can be colored by three colors but cannot be colored by two colors, so the graph shown has chromatic index three. By Vizing's theorem, the number of colors needed to edge color a simple graph is either its maximum degree or . For some graphs, such as bipartite graphs and high-degree planar graphs, the nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Arboricity
The arboricity of an undirected graph is the minimum number of forests into which its edges can be partitioned. Equivalently it is the minimum number of spanning forests needed to cover all the edges of the graph. The Nash-Williams theorem provides necessary and sufficient conditions for when a graph is ''k''-arboric. Example The figure shows the complete bipartite graph ''K''4,4, with the colors indicating a partition of its edges into three forests. ''K''4,4 cannot be partitioned into fewer forests, because any forest on its eight vertices has at most seven edges, while the overall graph has sixteen edges, more than double the number of edges in a single forest. Therefore, the arboricity of ''K''4,4 is three. Arboricity as a measure of density The arboricity of a graph is a measure of how dense the graph is: graphs with many edges have high arboricity, and graphs with high arboricity must have a dense subgraph. In more detail, as any n-vertex forest has at most n − 1 edges ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Complete Bipartite Graph
In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set..Electronic edition page 17. Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete bipartite graphs were already printed as early as 1669, in connection with an edition of the works of Ramon Llull edited by Athanasius Kircher. Llull himself had made similar drawings of complete graphs three centuries earlier.. Definition A complete bipartite graph is a graph whose vertices can be partitioned into two subsets and such that no edge has both endpoints in the same subset, and every possible edge that could connect vertices in different subsets is part of the graph. That is, it is a bipartite graph such that for every two vertices and, is an edge in . A complete bipartite graph w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Wheel Graph
In graph theory, a wheel graph is a graph formed by connecting a single universal vertex to all vertices of a cycle. A wheel graph with vertices can also be defined as the 1-skeleton of an pyramid. Some authors write to denote a wheel graph with vertices (); other authors instead use to denote a wheel graph with vertices (), which is formed by connecting a single vertex to all vertices of a cycle of length . The former notation is used in the rest of this article and in the table on the right. Edge Set would be the edge set of a wheel graph with vertex set in which the vertex 1 is a universal vertex. Properties Wheel graphs are planar graphs, and have a unique planar embedding. More specifically, every wheel graph is a Halin graph. They are self-dual: the planar dual of any wheel graph is an isomorphic graph. Every maximal planar graph, other than ''K''4 = ''W''4, contains as a subgraph either ''W''5 or ''W''6. There is always a Hamiltonian cycle in the wheel graph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
