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When modelling relations between two different classes of objects, bipartite graphs very often arise naturally. For instance, a graph of football players and clubs, with an edge between a player and a club if the player has played for that club, is a natural example of an affiliation network, a type of bipartite graph used in social network analysis.
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]
The Ruzsa–Szemerédi problem asks for the answer to these equivalent questions. To convert the bipartite graph induced matching problem into the unique triangle problem, add a third set of vertices to the graph, one for each induced matching, and add edges from vertices and of the bipartite graph to vertex in this third set whenever bipartite ...
A bipartite graph B = (X,Y,E) is chordal bipartite if and only if every induced subgraph of B has a maximum X-neighborhood ordering and a maximum Y-neighborhood ordering. [5] Various results describe the relationship between chordal bipartite graphs and totally balanced neighborhood hypergraphs of bipartite graphs. [6]
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 ...
The Folkman graph is the smallest undirected graph that is edge-transitive and regular, but not vertex-transitive. [6] Such graphs are called semi-symmetric graphs and were first studied by Folkman in 1967 who discovered the graph on 20 vertices that now is named after him. [2] Like all semi-symmetric graphs, the Folkman graph is bipartite. Its ...
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).
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.