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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 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 ...
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 ...
Many triangle-free graphs are not bipartite, for example any cycle graph C n for odd n > 3. By Turán's theorem, the n-vertex triangle-free graph with the maximum number of edges is a complete bipartite graph in which the numbers of vertices on each side of the bipartition are as equal as possible.
In the mathematical field of graph theory, the Folkman graph is a 4-regular graph with 20 vertices and 40 edges. It is a regular bipartite graph with symmetries taking every edge to every other edge, but the two sides of its bipartition are not symmetric with each other, making it the smallest possible semi-symmetric graph. [1]
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 [6] there is a different decomposition of a bipartite graph, which is asymmetric - it distinguishes between vertices in one side of the graph and the vertices on the other side. It can be used to find a maximum-cardinality envy-free matching in an unweighted bipartite graph, as well as a minimum-cost maximum-cardinality matching in a ...
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