Finding Smooth Graphs with Small Independence Numbers

In this paper we formulate an algorithm for finding smooth graphs with small independence numbers. To this end we formalize a family of satisfaction problems and propose a branch-and-bound-based approach for solving them. Strong bounds are obtained by exploiting graph-theoretic aspects including new results obtained in cooperation with leading graph theorists. Based on a partial solution we derive a lower bound by computing an independent set on a partial graph and finding a lower bound on the size of possible extensions. The algorithm is used to test conjectured lower bounds on the independence numbers of smooth graphs and some subclasses of smooth graphs. In particular for the whole class of smooth graphs we test the lower bound of 2n/7 for all smooth graphs with at least n ≥ 12 vertices and can proof the correctness for all 12 ≤ n ≤ 24. Furthermore, we apply the algorithm on different subclasses, such as all triangle free smooth graphs.