On Point-Color-Symmetric Graphs and Groups with Point-Color-Symmetric Picture Representation

Date of Award


Document Type


Degree Name

Master of Science in Mathematics



First Advisor

Reginaldo M. Marcelo, PhD


In 1979, Chen and Teh introduced the concept of point-color-symmetric (PCS) graphs and determined a necessary and sufficient condition for a graph to be PCS. Extending this, Marcelo et al. in 1994 defined conditions for a PCS graph to be a PCS picture representation (PPR) for a group and also determined a necessary and sufficient condition for a group to have a PPR. However, the determination of all groups with a PPR is not yet complete. This work determines necessary and sufficient conditions for the circulant graph C(n : i) to have a PCS edge coloring and a sufficient condition for C(n : i, j) to have a PCS edge coloring. A necessary and sufficient condition for the complete bipartite graph Km,n to have a PCS edge coloring is also shown. In addition, it is shown that for n > 1, the n-dimensional hypercube graph Qn has a PCS edge coloring with n colors, and that no wheel graph has a PCS edge coloring. Moreover, we show that for n > 2, the dihedral group Dn of order 2n has a PPR with two colors and a PPR with n colors and that the underlying graphs of these PPRs are C2n and Kn,n. Extending this to a larger group, a characterization for groups of the form Dn × Z2 to have a PPR is also determined. Boolean groups, that is, groups of the form Z n 2 are also shown to have a PPR that is precisely Qn. Finally, using the computational software GAP, we determine some finite Coxeter groups that have a PPR.

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