homomorphisms · 2019. 12. 7. · proof. assume that we are given homomorphisms f z -+ y. the map ....

Homomorphisms Lemma The chromatic number of a graph X is the least integer r such that there is a homomorphism from X to K r .

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Homomorphisms

Lemma The chromatic number of a graph X is the least integer r such that there is a homomorphism from X to Kr .

Core

Lemma 6.2.1 Let X and Y be cores. Then X and Yare homomorphically equivalent il and only il they are isomorphic.

Page 3: Homomorphisms · 2019. 12. 7. · Proof. Assume that we are given homomorphisms f Z -+ Y. The map . Z X and g . is readily seen to be a homomorphism from Z to X x Y. Clearly, PX 0

Lemma 6.2.2 Every graph has a core, which is an induced subgraph and is unique up to isomorphism.

Lemma 6.2.3 Two graphs X and Y are homomorphically equivalent if and only if their cores are isomorphic.

Page 4: Homomorphisms · 2019. 12. 7. · Proof. Assume that we are given homomorphisms f Z -+ Y. The map . Z X and g . is readily seen to be a homomorphism from Z to X x Y. Clearly, PX 0

Products

Page 5: Homomorphisms · 2019. 12. 7. · Proof. Assume that we are given homomorphisms f Z -+ Y. The map . Z X and g . is readily seen to be a homomorphism from Z to X x Y. Clearly, PX 0

Page 6: Homomorphisms · 2019. 12. 7. · Proof. Assume that we are given homomorphisms f Z -+ Y. The map . Z X and g . is readily seen to be a homomorphism from Z to X x Y. Clearly, PX 0

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Graph homomorphisms: