Journal of the Ramanujan Mathematical Society

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The Cyclomatic Number of Connected Graphs without Solvable Orbits


Affiliations
1 Institute of Mathematics, Eotvos University, Pazmany P.Setany 1/C, H–1117 Budapest, Hungary
2 Department of Mathematics, Imperial College, 180 Queen’s Gate, London SW7 2AZ, United Kingdom
     

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We study the combinatorics of constructing non-singular geometrically irreducible projective curves that do not admit rational points over finite solvable extensions of the base field. A graph is without solvable orbits if its group of automorphisms acts on each of its orbits through a non-solvable quotient. We prove that there is a connected graph without solvable orbits of cyclomatic number c if and only if c is equal to 6, 8, 10, 11, 15, 16, 19, 20, 21, 22, or is at least 24. For these numbers there exist smooth geometrically irreducible projective curves without solvable points whose genus equals c.
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  • The Cyclomatic Number of Connected Graphs without Solvable Orbits

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Authors

Gyula Karolyi
Institute of Mathematics, Eotvos University, Pazmany P.Setany 1/C, H–1117 Budapest, Hungary
Ambrus Pal
Department of Mathematics, Imperial College, 180 Queen’s Gate, London SW7 2AZ, United Kingdom

Abstract


We study the combinatorics of constructing non-singular geometrically irreducible projective curves that do not admit rational points over finite solvable extensions of the base field. A graph is without solvable orbits if its group of automorphisms acts on each of its orbits through a non-solvable quotient. We prove that there is a connected graph without solvable orbits of cyclomatic number c if and only if c is equal to 6, 8, 10, 11, 15, 16, 19, 20, 21, 22, or is at least 24. For these numbers there exist smooth geometrically irreducible projective curves without solvable points whose genus equals c.