Almost self-complementary circulant graphs

An almost self-complementary graph is a graph isomorphic to its complement minus a 1-factor. An almost self-complementary circulant graph is called cyclically almost self-complementary if it has an isomorphic almost complement with the same regular cyclic subgroup of the automorphism group. In this paper we prove that a cyclically almost self-complementary circulant of order 2n exists if and only if every prime divisor of n is congruent to 1 modulo 4, thus extending the known result on the existence of self-complementary circulants. We also describe the structure of cyclically almost self-complementary circulants and the action of their automorphism groups. Finally, we exhibit a class of almost self-complementary Cayley graphs on a dihedral group that are isomorphic to cyclically almost self-complementary circulants.

[1]  J. Dixon,et al.  Permutation Groups , 1996 .

[2]  W. Burnside,et al.  On some Properties of Groups of Odd Order , 1900 .

[3]  Gerhard Ringel.  Selbstkomplementäre Graphen , 1963 .

[4]  D. A. Suprunenko.  Self-complementary graphs , 1985 .

[5]  Reinhard Pöschel,et al.  Non-Cayley-isomorphic self-complementary circulant graphs , 2000 .

[6]  Edward Dobson,et al.  On the Cayley isomorphism problem , 2002, Discret. Math..

[7]  Edward Dobson,et al.  Isomorphism problem for Cayley graphs of Zp3 , 1995, Discret. Math..

[8]  P. Cameron.  FINITE PERMUTATION GROUPS AND FINITE SIMPLE GROUPS , 1981 .

[9]  Brian Alspach,et al.  A Construction for Vertex-Transitive Graphs , 1982, Canadian Journal of Mathematics.

[10]  S. B. Rao.  On regular and strongly-regular self-complementary graphs , 1985, Discret. Math..

[11]  László Babai,et al.  Isomorphism problem for a class of point-symmetric structures , 1977 .

[12]  Alexander Rosa,et al.  The Existence of Selfcomplementary Circulant Graphs , 1996, Eur. J. Comb..

[13]  Cai Heng Li,et al.  Constructions of Self-complementary Circulants with No Multiplicative Isomorphisms , 2001, Eur. J. Comb..

[14]  Joy Morris,et al.  Self-complementary circulant graphs , 1999, Ars Comb..

[15]  H. Wielandt,et al.  Finite Permutation Groups , 1964 .

[16]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[17]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[18]  Mikhail Muzychuk.  On Sylow Subgraphs of Vertex-Transitive Self-Complementary Graphs , 1999 .

[19]  Mikhail E. Muzychuk.  On Ádám's conjecture for circulant graphs , 1997, Discret. Math..

[20]  Peter Frankl,et al.  Isomorphisms of Cayley graphs. II , 1979 .

[21]  Mikhail E. Muzychuk,et al.  On Ádám's conjecture for circulant graphs , 1997, Discret. Math..