On the Graph of the Pedigree Polytope
  Pedigree polytopes are extensions of the classical Symmetric Traveling Salesman Problem polytopes whose graphs (1-skeletons) contain the TSP polytope graphs as spanning subgraphs. While deciding adjacency of vertices in TSP polytopes is coNP-complete, Arthanari has given a combinatorial (polynomially decidable) characterization of adjacency in Pedigree polytopes. Based on this characterization, we study the graphs of Pedigree polytopes asymptotically, for large numbers of cities. Unlike TSP polytope graphs, which are vertex transitive, Pedigree graphs are not even regular. Using an "adjacency game" to handle Arthanari's intricate inductive characterization of adjacency, we prove that the minimum degree is asymptotically equal to the number of vertices, i.e., the graph is "asymptotically almost complete". 
Abdullah Makkeh and Mozhgan Pourmoradnasseri and Dirk Oliver Theis
cs.DM math.CO