By Kasimierz Kuratowski (auth.), M. Borowiecki, John W. Kennedy, Maciej M. Sysło (eds.)
Read or Download Graph Theory: Proceedings of a Conference held in Łagów, Poland, February 10–13, 1981 PDF
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Additional info for Graph Theory: Proceedings of a Conference held in Łagów, Poland, February 10–13, 1981
Sample text
Now suppose that G and H have the property phism of: G onto a subgraph Case (I). V(H ) as an arbitrary (i ) (I) or (2a) and let qb0 be an isomor- of H in (I) or G I onto H I in (2a). for every ucV(H'), vcV(H"), mapping (H',H",F'), where 0 = H'. Let us such that IF-1(¢-~1(u),v)I = IF,-1(u,v) I, and (ii ) for every p~V(G), qcV(H~), Then a nonnatural IF-1(¢o(p),¢o1(q))I = IF-1(p,q)l . %m ¢ of (G,H,F) can be defined by ¢(u) = Case (2a). U for ueV(Hv'), t0(u) for uEV(G), q~0l(u) for ucV(H'). V(H) and a nonnatural (G,H,F) in similar way to Case (I).
Fn(G,H,F)/F~(G,H,F ) = S I or S 2 (symmetric groups). 2. If 41 and 42 are automorphisms of G and H, respectively, then there exists a positive natural automorphism 4 of (G,H,F) with 4/V(G) = 41 and 4/V(H) = 42 if and only if for every pair ucV(G), vcV(H), iF-l(u,v)l = IS1(41(u),,2(v))l. 48 If ~ exists, then it is uniquely determined by ~I and %2 and all positive natura~ automorphisms are generated in this way. P roof. First, let qb be a positive natural automorphism of (G,H,F). Then % induces two automorphisms %1 and %2 of G and H, respectively, by %1 = %/V(G) and %2 = %/V(H).
Let G I be isomorphic automorphism of to a subgraph H I of H. Then 54 the graph H has a form (HI,H2,F2). V(H) (i) for every ueV(HI) , vcV(H2) , fF-1(ti1(u),v)l such that = IF21(u,v)F and (ii) for every pcV(G1) , qEV(HI) , IF-1(¢o(p),¢o1(q))I = IF-1(p,q)I, and (iii) for every rEV(GI) , seV(G2) , IF-1(¢0(r),s)I = IF~1(r,s)I. 1. V(H) is one-to-one and neither G nor H is a mapping graph ~I,G2,FI) such that F I is one-to-one, then the mapping graph (G,H,F) has only natural automorphisms. 2. Let f: V' ~V(H), V'~V(G).