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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 ...
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]
In graph-theoretic mathematics, a biregular graph [1] or semiregular bipartite graph [2] is a bipartite graph = (,,) for which every two vertices on the same side of the given bipartition have the same degree as each other.
A bipartite graph with 4 vertices on each side, 13 edges, and no , subgraph, and an equivalent set of 13 points in a 4 × 4 grid, showing that (;).. The number (;) asks for the maximum number of edges in a bipartite graph with vertices on each side that has no 4-cycle (its girth is six or more).
The bipartite realization problem is equivalent to the question, if there exists a labeled bipartite subgraph of a complete bipartite graph to a given degree sequence. The hitchcock problem asks for such a subgraph minimizing the sum of the costs on each edge which are given for the complete bipartite graph.
Mathematically, this problem can be formalized as asking for a graph drawing of a complete bipartite graph, whose vertices represent kilns and storage sites, and whose edges represent the tracks from each kiln to each storage site. The graph should be drawn in the plane with each vertex as a point, each edge as a curve connecting its two ...
An example of a bipartite graph, with a maximum matching (blue) and minimum vertex cover (red) both of size six. In the mathematical area of graph theory, Kőnig's theorem, proved by Dénes Kőnig (), describes an equivalence between the maximum matching problem and the minimum vertex cover problem in bipartite graphs.
In the mathematical field of graph theory, a convex bipartite graph is a bipartite graph with specific properties. A bipartite graph, (U ∪ V, E), is said to be convex over the vertex set U if U can be enumerated such that for all v ∈ V the vertices adjacent to v are consecutive. Convexity over V is defined analogously. A bipartite graph (U ...