Iterated claws have real-rooted genus polynomials


  • Jonathan L. Gross Columbia University, New York
  • Toufik Mansour University of Haifa, Israel
  • Thomas W. Tucker Colgate University, United States
  • David G. L. Wang Beijing Institute of Technology, China



Topological graph theory, graph genus polynomials, log-concavity, real-rootedness


We prove that the genus polynomials of the graphs called iterated claws are real-rooted. This continues our work directed toward the 25-year-old conjecture that the genus distribution of every graph is log-concave. We have previously established log-concavity for sequences of graphs constructed by iterative vertex-amalgamation or iterative edge-amalgamation of graphs that satisfy a commonly observable condition on their partitioned genus distributions, even though it had been proved previously that iterative amalgamation does not always preserve real-rootedness of the genus polynomial of the iterated graph. In this paper, the iterated topological operation is adding a claw, rather than vertex- or edge-amalgamation. Our analysis here illustrates some advantages of employing a matrix representation of the transposition of a set of productions.





GEMS 2013