Abstract. Let [Formula: see text] be a 2-connected [Formula: see text]-vertex graph, and let [Formula: see text] be the total number of [Formula: see text]-cliques in [Formula: see text]. Let [Formula: see text] and [Formula: see text] be integers. In this paper, we show that if [Formula: see text] has an edge [Formula: see text] which is not on any cycle of length at least [Formula: see text], then [Formula: see text], where [Formula: see text] and [Formula: see text]. This result settles a conjecture of Ma and Yuan and provides a clique version of a result of Fan [ J. Combin. Theory Ser. B, 49 (1990), pp. 151–180], and a result of Wang and Lv [ Discrete Math., 308 (2008), pp. 113–122]. As a direct corollary, if [Formula: see text], every edge of [Formula: see text] is covered by a cycle of length at least [Formula: see text].

For a graph G, let $p(G)$ and $c(G)$ denote the number of vertices in a longest path and a longest cycle in G, respectively. In this paper, we prove that if G is a 2-connected graph G on n vertices with $p(G)=p$, where $p\geq20$, and if G has more than $\frac{1}{2}(p-2)(n-7)+13$ edges, then $p(G)-c(G)\leq1$, which implies that every longest cycle in G is a dominating cycle.