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In graph theory, the Graham–Pollak theorem states that the edges of an -vertex complete graph cannot be partitioned into fewer than complete bipartite graphs. [1] It was first published by Ronald Graham and Henry O. Pollak in two papers in 1971 and 1972 (crediting Hans Witsenhausen for a key lemma), in connection with an application to ...
The degree sequence of a bipartite graph is the pair of lists each containing the degrees of the two parts and . For example, the complete bipartite graph K 3,5 has degree sequence (,,), (,,,,). Isomorphic bipartite graphs have the same degree sequence. However, the degree sequence does not, in general, uniquely identify a bipartite graph; in ...
For a graph G, let χ(G) denote the chromatic number and Δ(G) the maximum degree of G.The list coloring number ch(G) satisfies the following properties.. ch(G) ≥ χ(G).A k-list-colorable graph must in particular have a list coloring when every vertex is assigned the same list of k colors, which corresponds to a usual k-coloring.
A stronger definition of bipartiteness is: a hypergraph is called bipartite if its vertex set V can be partitioned into two sets, X and Y, such that each hyperedge contains exactly one element of X. [2] [3] Every bipartite graph is also a bipartite hypergraph. Every bipartite hypergraph is 2-colorable, but bipartiteness is stronger than 2 ...
A complete bipartite graph K m,n has a maximum matching of size min{m,n}. A complete bipartite graph K n,n has a proper n-edge-coloring corresponding to a Latin square. [14] Every complete bipartite graph is a modular graph: every triple of vertices has a median that belongs to shortest paths between each pair of vertices. [15]
Some of the local methods assume that the graph admits a perfect matching; if this is not the case, then some of these methods might run forever. [1]: 3 A simple technical way to solve this problem is to extend the input graph to a complete bipartite graph, by adding artificial edges with very large weights. These weights should exceed the ...
The set of users together with the set of resources in the system naturally induces a bipartite graph, whose edges are permissions. Each biclique in this graph is a potential role, and the optimum solutions to the role mining problem are precisely the minimum biclique edge covers (Ene et al. 2008).
In graph theory, a split of an undirected graph is a cut whose cut-set forms a complete bipartite graph.A graph is prime if it has no splits. The splits of a graph can be collected into a tree-like structure called the split decomposition or join decomposition, which can be constructed in linear time.