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Homomorphisms of infinite bipartite graphs onto complete bipartite graphs. (English) Zbl 0543.05058
Let B be a bipartite graph on the vertex sets C, D. A homomorphism $$\phi$$ of B onto a complete bipartite graph $$K_{r,s}$$ is said to be bicomplete if $$\phi(x)=\phi(y)$$ only if either both x, y belong to C, or both x, y belong to D. For a connected bipartite graph B, the author defines the parameter $$\beta_ 0(B)$$ as the supremum of all values of min(r,s) for all complete bipartite graphs $$K_{r,s}$$ onto which B can be mapped by a bicomplete homomorphism. As the author points out, this parameter $$\beta_ 0(B)$$ is closely related to the bichromaticity $$\beta$$ (B) which was introduced for finite B by F. Harary, D. Hsu and Z. Miller [Theor. Appl. Graphs, Proc. Kalamazoo 1976, Lect. Notes Math. 642, 236-246 (1978; Zbl 0369.05022)]; however, for infinite B, $$\beta_ 0(B)$$ is more interesting than $$\beta$$ (B) since one can easily show that $$\beta$$ (B) equals the cardinality of the vertex set of B. For infinite B, the author proves two theorems on $$\beta_ 0(B)$$ which relate $$\beta_ 0(B)$$ to other parameters such as the supremum of the cardinalities of all matchings of B.
Reviewer: T.Andreae

##### MSC:
 05C99 Graph theory 05C15 Coloring of graphs and hypergraphs
##### Keywords:
bipartite graph; bicomplete homomorphism; bichromaticity
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##### References:
 [1] Harary F., Hsu D., Miller Z.: The bichromaticity of a tree. Theory and Applocations of Graphs. Proc. Michigan 1976. [2] Ore O.: Theory of Graphs. Providence 1962. · Zbl 0105.35401
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