# Download e-book for iPad: (2,k)-Factor-Critical Graphs and Toughness by Cai M.-C., Favaron O., Li H.

By Cai M.-C., Favaron O., Li H.

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N − 1} then l is a semilinear isomorphism. Proof. 8. 2. Let F and E be division rings. Suppose that F is a proper subring of E. Then for every natural m ≥ 2 and every k ∈ {1, . . , m − 1} there exists U ∈ Gk (E m ) such that the dimension of the subspace U ∩ F m in the vector space F m is less than k. page 22 March 19, 2015 14:43 BC: 9465 - Geometry of Semilinear Embeddings Semilinear mappings main 23 Proof. 1, the mapping of Gk (F m ) to Gk (E m ) induced by the identity embedding of F m in E m is non-surjective.

2. Let f : G1 (V ) → G1 (V ′ ) be a mapping satisfying the following condition: if P, P1 , P2 ∈ G1 (V ) then P ⊂ P1 + P2 =⇒ f (P ) ⊂ f (P1 ) + f (P2 ). If the image f (G1 (V )) is not contained in a line of ΠV ′ then f is induced by a semilinear injection of V to V ′ . page 40 March 19, 2015 14:43 BC: 9465 - Geometry of Semilinear Embeddings Projective Geometry and linear codes main 41 The proof of this statement is a modification of the proof of the classical Fundamental Theorem of Projective Geometry [Artin (1957); Baer (1952)].

If p ∈ s then f (s) coincides with l. e. the restriction of f to s is not injective or constant. The latter means that f cannot be induced by a semilinear transformation of the associated vector space. 2 implies the existence of a semilinear injection l : V → V ′ such that f = (l)1 . Since f transfers any triple of non-collinear points to a triple of non-collinear points, l is a semilinear 3-embedding. 8 and get the classical version of the Fundamental Theorem of Projective Geometry. 2. Every collineation of ΠV to ΠV ′ is induced by a semilinear isomorphism of V to V ′ .

### (2,k)-Factor-Critical Graphs and Toughness by Cai M.-C., Favaron O., Li H.

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