## Extremal Problems in Graph Homomorphisms and Vertex Identifications |

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### Contents

Bichromaticity | 5 |

Applications of Bichromaticity | 25 |

Analogues of a problem of Zarankiewicz | 30 |

4 other sections not shown

### Common terms and phrases

achromatic number acyclic graph acyclic morphic image adjacent vertices bichromaticity bipartite graph bipartition M,N choosing chromatic clique color columns complete bipartite graphs connected components contradiction corresponding diam(G Ferrars matrices fl(G G of order graph G GRAPH HOMOMORPHISMS Graph Theory graphs of order greedy algorithm Harary homomor image of G induced subgraph inductive hypothesis integer isolated vertices k-colorable least integer Lemma lower bound majority side vertex matrix maximum minimum number minimum order minimum total weight morphism from G neighbors normal homomorphic image number of edges number of vertices obtain odd degree odd-morphic image partition of V(G partition problems pendant phic image preimages present in G produce G-V produces G Proof proper homomorphism proposition holds rows spanning forest submatrix Suppose Theorem Theorem 4.2 up-diagonal upper bound vector in Q VERTEX IDENTIFICATIONS vertex set vertices of G vertices of odd W.T. Tutte weight-3 weight-4 vectors XK(G