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

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Is not totally disconnected, then , if and only if: G(G~[G~]) Z G(G1) [ f i ( G 2 ) l (i) if there are two vertices in same neighborhood, then G2 G1 with the is connected. and (ii) if there are two vertices in G1 with the same closed neighborhood, then G2 is connected. We are now able to list the automorphism groups for several common families of graphs. The Automorphism Group of a Graph 20 3-4. Chapt. 3 Graphs with a Given Automorphism Group Every finite group is the automorphism group of some Thm.

The result is a 2-cell imbedding of a connected pseudograph in with parameters p' , q ' , and r' ( s a y ) . But 'k-1 Proof I ... p' = p + 2n q' = q + + r' = r 3n n + 2. Sect. 5-4 Two Applications 47 Thus, by the inductive assumption, Cor. 5-15. Let G in Sk p - be a connected graph, with a 2-cell imbedding , with the parameters p, q , and r; then q +r = 2 Proof: - 2k. The result is immediate, since any graph is also a pseudograph. # We have shown that the number p - q + r is invariant for k' for any 2-cell imbedding of any connected pseudograph; p - q + r = 2 - 2k, depending only on k.

By Theorem I t only remains t o c o n v e r t where 3 A t vertex P. (P. ' ) of l e n g t h 11 1 3 f o r t h e case k = 2 . i n t o t h e graph be a g e n e r a t i n q C,(l') h a v i n g t h e same a u t o m o r p h i s m g r o u p , G T h i s i s done as f o l l o w s : vi A b e a f i n i t e g r o u p , and l e t I' 4 Chapt. , C,(I') It is clear that features a r e incorporated G(G) G(CA(I')) # 1'. V. u.. 17 u v i j j F i g u r e 4-4 4-3 Properties I t i s c l e a r t h a t e v e r y Cayley c o l o r g r a p h i s b o t h r e g u l a r and c o n n e c t e d ( a s a g r a p h ) ; t h e c o n v e r s e i s not t r u e (see P r o b l e m 4 - 1 0 ) .

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

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