On cycles in intersection graphs of rings

Document Type : Research Paper


Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran.


‎Let $R$ be a commutative ring with non-zero identity. ‎We describe all $C_3$‎- ‎and $C_4$-free intersection graph of non-trivial ideals of $R$ as well as $C_n$-free intersection graph when $R$ is a reduced ring. ‎Also, ‎we shall describe all complete, ‎regular and $n$-claw-free intersection graphs. ‎Finally, ‎we shall prove that almost all Artin rings $R$ have Hamiltonian intersection graphs. ‎We show that such graphs are indeed pancyclic.


Main Subjects

A. Akbari, R. Nikandish and M. J. Nikmehr, Some results on the intersection graphs of ideals of rings, J. Algebra Appl. 12 (2013), no. 4, 13 pages.
J. Bos_ak, The graphs of semigroups, Theory of Graphs and Application, 119--125, Publ. House Czechoslovak Acad. Sci., Prague, 1964.
I. Chakrabarty, S. Ghosh, T. K. Mukherjee and M. K. Sen, Intersection graphs of ideals of rings, Discrete Math. 309 (2009), no. 17, 5381--5392.
B. Cs_ak_eany and G. Poll_ak, The graph of subgroups of a finite group, Czechoslovak Math. J. 19 (1969) 241--247.
M. Herzog, P. Longobardi and M. Maj, On a graph related to the maximal subgroups of a group, Bull. Aust. Math. Soc. 81 (2010), no. 2, 317--328.
S. H. Jafari and N. Jafari Rad, Domination in the intersection graphs of rings and modules, Ital. J. Pure Appl. Math. 28 (2011) 19--22.
S. H. Jafari and N. Jafari Rad, Planarity of intersection graphs of ideals of rings, Int. Electron. J. Algebra 8 (2010) 161--166.
B. R. Macdonald, Finite Rings with Identity, Marcel Dekker, Inc., New York, 1974.
T. A. McKee and F. R. McMorris, Topic in Intersection Graph Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1999.
R. Shen, Intersection graphs of subgroups of finite groups, Czech. Math. J. 60(135) (2010), no. 4, 945--950.
B. Zelinka, Intersection graphs of finite abelian groups, Czechoslovak Math. J. 25 (1975) 171--174.