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The Shrikhande graph shares these parameters with exactly one other graph, the 4×4 rook's graph, i.e., the line graph L(K 4,4) of the complete bipartite graph K 4,4. The latter graph is the only line graph L ( K n,n ) for which the strong regularity parameters do not determine that graph uniquely but are shared with a different graph, namely ...
In graph theory, a part of mathematics, a k-partite graph is a graph whose vertices are (or can be) partitioned into k different independent sets. Equivalently, it is a graph that can be colored with k colors, so that no two endpoints of an edge have the same color. When k = 2 these are the bipartite graphs, and when k = 3 they are called the ...
A global characterization of Krausz type for the line graphs of k-uniform linear hypergraphs for any k ≥ 3 was given by Naik, Rao, Shrikhande, and Singhi. [6] At the same time, they found a finite list of forbidden induced subgraphs for linear 3-uniform hypergraphs with minimum vertex degree at least 69.
Sharadchandra Shankar Shrikhande (19 October 1917 – 21 April 2020) was an Indian mathematician with notable achievements in combinatorial mathematics.He was notable for his breakthrough work along with R. C. Bose and E. T. Parker in their disproof of the famous conjecture made by Leonhard Euler dated 1782 that there do not exist two mutually orthogonal latin squares of order 4n + 2 for any n ...
These graphs include the complete graph K 6, the Petersen graph, the graph formed by removing an edge from the complete bipartite graph K 4,4, and the complete tripartite graph K 3,3,1. Every planar graph has a flat and linkless embedding: simply embed the graph into a plane and embed the plane into space. If a graph is planar, this is the only ...
Every bipartite graph G = (X+Y, E) is 2-colorable: each edge contains exactly one vertex of X and one vertex of Y, so e.g. X can be colored blue and Y can be colored yellow and no edge is monochromatic. The property of 2-colorability was first introduced by Felix Bernstein in the context of set families; [1] therefore it is also called Property B.
[1] [2] The 7-page book graph of this type provides an example of a graph with no harmonious labeling. [2] A second type, which might be called a triangular book, is the complete tripartite graph K 1,1,p. It is a graph consisting of triangles sharing a common edge. [3] A book of this type is a split graph.
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.