Forbidden subgraphs for longest cycles to contain vertices with large degrees

Abstract

a b s t r a c tLet G be a graph. For a given graph H, we say that G is H-free if G contains no copies of H as an induced subgraph. Suppose that G is 2-connected, has n vertices, and α is a real number with 0 ≤ α ≤ 1. In this paper, we characterize the connected graphs R such that G being R-free implies that every longest cycle of G passes through all vertices with degree at least αn + O(1) in G.

Large degree vertices in longest cycles of graphs II

“…Note that if G is 2-connected and M has at least 2 vertices, then every component of G M is locally 2-connected to M . We will also use the following lemmas from [2] and [10].…”