ISOMORPHISM CLASSES OF CAYLEY PERMUTATION GRAPHS

Abstract

In this paper, we study the isomorphism problem of Cayley permutation graphs. We obtain a necessary and sufficient condition that two Cayley permutation graphs are isomrphic by a direction-preserving and color-preserving (positive/negative) natural isomorphism. The result says that if a graph G is the Cayley graph for a group $\Gamma$ then the number of direction-preserving and color-preserving positive natural isomorphism classes of Cayley permutation graphs of G is the number of double cosets of $\Gamma^\ell$ in $S_\Gamma$, where $S_\Gamma$ is the group of permutations on the elements of $\Gamma and \Gamma^\ell$ is the group of left translations by the elements of $\Gamma$. We obtain the number of the isomorphism classes by counting the double cosets.

Keywords

• Cayley permutation graph;
• isomorphism

File

Download PDF

References

1. Ann. Inst. Henri Poincare III v.4 Planar permutation graphs G. Chartrand;F. Harary
2. Topological graph theory J. L. Gross;T. W. Tucker
3. The many facets of graph theory, Lecture Notes in Mathematics 110 On classes of graphs dened by special cutsets of lines S. Hedetniemi
4. Combinatorial Mathematics III, Lecture Notes in Mathematics 452 Some problems in permutation graphs D. A. Holton;K. C. Stacey
5. Discrete Math. v.105 Isomorphism classes of cycle permutation graphs J. H. Kwak;J. Lee
6. Discrete Math. v.51 On cycle permutation graphs R. Ringeisen
7. Introduction to Abstract Algebra L. Shapiro
8. Graphs Combin. v.4 On natural isomorphisms of cycle permutation graphs S. Stueckle