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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 ...
Any regular bipartite graph. [1] Hall's marriage theorem can be used to show that a k-regular bipartite graph contains a perfect matching. One can then remove the perfect matching to obtain a (k − 1)-regular bipartite graph, and apply the same reasoning repeatedly. Any complete graph with an even number of nodes (see below). [2]
Every tree is a bipartite graph. A graph is bipartite if and only if it contains no cycles of odd length. Since a tree contains no cycles at all, it is bipartite. Every tree with only countably many vertices is a planar graph. Every connected graph G admits a spanning tree, which is a tree that contains every vertex of G and whose edges are ...
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 theory, the graph removal lemma states that when a graph contains few copies of a given subgraph, then all of the copies can be eliminated by removing a small number of edges. [1] The special case in which the subgraph is a triangle is known as the triangle removal lemma .
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).
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A graph is said to be maximal-clique irreducible if every maximal clique has an edge that belongs to no other maximal clique, and hereditary maximal-clique irreducible if the same property is true for every induced subgraph. [3] Hereditary maximal-clique irreducible graphs include triangle-free graphs, bipartite graphs, and interval graphs.