Independence Number of 2-Factor-Plus-Triangles Graphs

A 2-factor-plus-triangles graph is the union of two 2-regular graphs G1 and G2 with the same vertices, such that G2 consists of disjoint triangles. Let G be the family of such graphs. These include the famous “cycle-plus-triangles” graphs shown to be 3-choosable by Fleischner and Stiebitz. The independence ratio of a graph in G may be less than 1/3; but achieving the minimum value 1/4 requires each component to be isomorphic to a single 12-vertex graph. Nevertheless, G contains infinitely many connected graphs with independence ratio less than 4/15. For each odd g there are infinitely many connected graphs in G such that G1 has girth g and the independence ratio of G is less than 1/3. Also, when 12 divides n (and n 6= 12) there is an n-vertex graph in G such that G1 has girth n/2 and G is not 3-colorable. Finally, unions of two graphs whose components have at most s vertices are s-choosable.