Local Connectivity, Local Degree Conditions, some Forbidden Induced Subgraphs, and Cycle Extendability
Abstract
The research in the present paper was motivated by the conjecture of Ryj\'acek that every locally connected graph is weakly pancyclic. For a connected locally connected graph G of order at least 3, our results are as follows: If G is (K1+(K1 K2))-free, then G is weakly pancyclic. If G is (K1+(K1 K2))-free, then G is fully cycle extendable if and only if 2δ(G)≥ n(G). If G is \ K1+K1+K3,K1+P4\-free or \ K1+K1+K3,K1+(K1 P3)\-free, then G is fully cycle extendable. If G is distinct from K1+K1+K3 and \ K1+P4,K1,4,K2+(K1 K2)\-free, then G is fully cycle extendable. Furthermore, if G is a connected graph of order at least 3 such that |NG(u) NG(v) NG(w)|>|NG(u) (NG[v] NG[w])| for every induced path vuw of order 3 in G, then G is fully cycle extendable, which implies that every connected locally Ore or locally Dirac graph of order at least 3 is fully cycle extendable.