Characterizing forbidden pairs for relative length of longest paths and cycles

For a set H of connected graphs, a graph G is said to be H-free if G does not contain any member of H as an induced subgraph. When |H|=2, H is called a forbidden pair. In this paper, we completely characterize the forbidden pairs H such that every 2-connected H-free graph G satisfies p(G)−c(G)≤1, where p(G) and c(G) denote the order of a longest path and a longest cycle of G, respectively. This strengthens some result of Chiba et al. (2015) [7]. Furthermore, we investigate the forbidden pairs needed to guarantee a 2-connected H-free graph G satisfying p(G)−c(G)≤k for any positive integer k. Meanwhile, we determine the forbidden pairs H such that every 2-connected H-free graph G satisfies c(G)≥n−k for any positive integer k. These results extend the work of Faudree and Gould (1997) [12] on Hamilton cycles.