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Tanner proved the following bounds Let be the rate of the resulting linear code, let the degree of the digit nodes be and the degree of the subcode nodes be .If each subcode node is associated with a linear code (n,k) with rate r = k/n, then the rate of the code is bounded by
The conjecture was first proven for bipartite, cubic, bridgeless graphs by Voorhoeve (1979), later for planar, cubic, bridgeless graphs by Chudnovsky & Seymour (2012). The general case was settled by Esperet et al. (2011) , where it was shown that every cubic, bridgeless graph contains at least 2 n / 3656 {\displaystyle 2^{n/3656}} perfect ...
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.
One of these subclasses was the family of claw-free graphs: it was discovered by several authors that claw-free graphs without odd cycles and odd holes are perfect. Perfect claw-free graphs may be recognized in polynomial time. In a perfect claw-free graph, the neighborhood of any vertex forms the complement of a bipartite graph.
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 graph theory, a branch of mathematics, the Herschel graph is a bipartite undirected graph with 11 vertices and 18 edges. It is a polyhedral graph (the graph of a convex polyhedron), and is the smallest polyhedral graph that does not have a Hamiltonian cycle, a cycle passing through all its vertices.
To prove Dilworth's theorem for a partial order S with n elements, using Kőnig's theorem, define a bipartite graph G = (U,V,E) where U = V = S and where (u,v) is an edge in G when u < v in S. By Kőnig's theorem, there exists a matching M in G , and a set of vertices C in G , such that each edge in the graph contains at least one vertex in C ...
Also, any induced subgraph of a bipartite graph remains bipartite. Therefore, bipartite graphs are perfect. In n-vertex bipartite graphs, a minimum clique cover takes the form of a maximum matching together with an additional clique for every unmatched vertex, with size n − M, where M is the cardinality of the