Stability in the Erdős–Gallai Theorem on cycles and paths, II

Abstract

The Erdős-Gallai Theorem states that for k ≥ 3, any n-vertex graph with no cycle of length at least k has at most 1 2 (k −1)(n−1) edges. A stronger version of the Erdős-Gallai Theorem was given by Kopylov: If G is a 2-connected n-vertex graph with no cycle of length at least k, then e(G) ≤ max{h(n, k, 2), h(n, k, k−1 2

Stability Results on the Circumference of a Graph

“…In this subsection, we give a sketch of the proof of Theorem 1.9, which we emphasize is quite different from the existing ones in [12,14,15].…”

“…is a nontrivial star forest 4 , (A, B) is complete bipartite, and there exist two vertices a, b ∈ A such that every star S in G[Y ] is {a, b}feasible: that is, N G (S) = {a, b} and if |V (S)| ≥ 3, then all leaves of S have degree 2 in G and have a common neighbor in {a, b}. Theorem 1.6 (Füredi, Kostochka, Luo, and Verstraëte [14]). Let G be a 2-connected graph on n vertices with circumference c, where 10 ≤ c ≤ n−1.…”

The number of cliques in graphs covered by long cycles

“…A stability result of the Theorem 1.2 is obtained Ma and Yuan [11], which also can be viewed as the clique version of a stability result of Theorem 1.1 given by Füredi, Kostochka and Verstraëte [5].…”

Anti-Ramsey numbers for paths

“…The extremal results of an n-vertex graph proved by stability results usually need n to be sufficiently large. Our proof of Theorem 1 bases on a recently result of Füredi, Kostochka, Luo and Verstraëte [7,8] holding for graphs with arbitrary number of vertices. Hence, we can apply the stability results to determine the exactly anti-Ramsey number for paths.…”