Stability Results on the Circumference of a Graph In this paper, we extend and refine previous Turantype results on graphs with a given circumference. Let Wn, k, c be the graph obtained from a clique Kc xe2x88x92 k  1 by adding n xe2x88x92 (c xe2x88x92 k 1) isolated vertices each joined to the same k vertices of the clique, and let f(n, k, c)  e(Wn, k, c). Improving a celebrated theorem of Erdxc5x91s and Gallai 8, Kopylov 18 proved that for c   max leftfleft(n,,3,,c right),;fleft(n,;leftlfloor c over 2 rightrfloor  1,;c right) right), then either G is a subgraph of Wn,2,c or (W_n,leftlfloor c over 2 rightrfloor ,c), or c is odd and G is a subgraph of a member of two wellcharacterized families which we define as xcfx87n,c and xcexb3n,c.

Many triangles with few edges Extremal problems concerning the number of independent sets or complete subgraphs in a graph have been well studied in recent years. Cutler and Radcliffe proved that among graphs with n vertices and maximum degree at most r, where n  a(r  1)  b and 0 xe2x89xa4 b xe2x89xa4 r, aKr1 xe2x88xaa Kb has the maximum number of complete subgraphs, answering a question of Galvin. Gan, Loh and Sudakov conjectured that aKr1 xe2x88xaaKb also maximizes the number of complete subgraphs Kt for each fixed size t xe2x89xa53, and proved this for a  1. Cutler and Radcliffe proved this conjecture for r xe2x89xa4 6. We investigate a variant of this problem where we fix the number of edges instead of the number of vertices. We prove that aKr1 xe2x88xaaC(b), where C(b) is the colex graph on b edges, maximizes the number of triangles among graphs with m edges and any fixed maximum degree r xe2x89xa4 8, where m  a(r1 2 )  b and 0 xe2x89xa4 b u003c (r1 2 ). Mathematics Subject Classifications: 05.