Recursive Algorithms for Generation of Planar Graphs

Abstract

In this thesis we introduce recursive algorithms for generation of two families of plane graphs. These algorithms start with small graphs and iteratively convert them to larger graphs. The families studied in this thesis are k-angulations (plane graphs whose faces are of size k) and plane graphs with a given face size sequence. We also design a very fast method for canonical embedding and isomorphism rejection of plane graphs. Most graph generators like plantri generate graphs up to isomorphism of the embedding, however our method does the isomorphism checking up to the underlying graph while taking advantage of the planarity and embeddings to speed up the computation. The next subject discussed in this thesis is a type of graph called hypohamiltonian in which after removing each vertex from the graph, there is a Hamiltonian cycle through all remaining vertices while the original graph does not have any such cycle. One of the problems in the literature since 1976 is to find the smallest planar hypohamiltonian graphs. The previous record by Weiner and Araya was a planar graph with 42 vertices. We improve this record by finding 25 planar hypohamiltonian graphs on 40 vertices while discovering many larger ones on 42 and 43 vertices. The final subject in the thesis is a family of molecules called fullerenes which are entirely composed of carbon atoms. The structure of fullerenes are 3-connected plane graphs with exactly 12 faces of size 5 and the rest of size 6. A famous conjecture regarding fullerenes, called face-spiral conjecture claims that the drawing of their graph can be unwound in a spiral manner starting from one face and circulating around that face until all faces are traversed. This conjecture is known to be incorrect and the smallest counterexample is made of 380 carbon atoms. We have extended this conjecture to families of 3-connected plane graphs with f3, f4 and f5 faces of size 3, 4, and 5 while the remaining faces have size 6 and found counterexample for all possible values of 〈 f3, f4, f5〉. We also found the smallest counterexamples for 11 of these families out of 19 possible cases.