Partitioning Sparse Graphs Into Triangles Relations to exact satisfiability and very fast exponential time algorithms

We consider the problem of partitioning bounded degree graphs into triangles. We show that this problem is polynomial time solvable on graphs of maximum degree three by giving a linear time algorithm. We also show that this problem becomes NP-complete on graphs of maximum degree four. Moreover, we show that there is no subexponential time algorithm for this problem on maximum degree four graphs unless the Exponential Time Hypothesis fails. However, the partition into triangles problem for graphs of maximum degree at most four is in many cases practically solvable as we give an algorithm for this problem that runs in O(1.0222n) time and linear space. We note that the running time of this algorithm does not involve any large hidden polynomial factors.