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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 complement graph of a complete graph is an empty graph. If the edges of a complete graph are each given an orientation, the resulting directed graph is called a tournament. K n can be decomposed into n trees T i such that T i has i vertices. [6] Ringel's conjecture asks if the complete graph K 2n+1 can be decomposed into copies of any tree ...
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 claw. In graph theory, an area of mathematics, a claw-free graph is a graph that does not have a claw as an induced subgraph.. A claw is another name for the complete bipartite graph, (that is, a star graph comprising three edges, three leaves, and a central vertex).
In graph theory, a star S k is the complete bipartite graph K 1,k : a tree with one internal node and k leaves (but no internal nodes and k + 1 leaves when k ≤ 1). Alternatively, some authors define S k to be the tree of order k with maximum diameter 2; in which case a star of k > 2 has k − 1 leaves. A star with 3 edges is called a claw.
A complete k-partite graph is a k-partite graph in which there is an edge between every pair of vertices from different independent sets. These graphs are described by notation with a capital letter K subscripted by a sequence of the sizes of each set in the partition.
For instance, if G = K m,n, the complete bipartite graph, then the second type of matching polynomial is related to the generalized Laguerre polynomial L n α (x) by the identity: , =! (). If G is the complete graph K n, then M G (x) is an Hermite polynomial:
Every complete graph is a comparability graph, the comparability graph of a total order. All acyclic orientations of a complete graph are transitive. Every bipartite graph is also a comparability graph. Orienting the edges of a bipartite graph from one side of the bipartition to the other results in a transitive orientation, corresponding to a ...