Stability in the Erdos--Gallai Theorem on cycles and paths

TL;DR

This paper establishes a stability version of the Erdős-Gallai Theorem for cycles and paths, showing that graphs near the extremal edge count either contain long cycles or have a specific structure after vertex removal.

Contribution

It proves a stability result for the Erdős-Gallai Theorem, characterizing the structure of graphs close to the extremal edge count for long cycles.

Findings

Graphs with more than h(n,k,t-1) edges contain long cycles or specific star forest structures.

The bound e(G) > h(n,k,t-1) is tight and improves upon Kopylov's bound by a linear term.

For k=2t+1≠7, such graphs are contained within the extremal family H_{n,k,t}.

Abstract

The Erd\H{o}s-Gallai Theorem states that for $k \geq 2$, every graph of average degree more than $k - 2$ contains a $k$-vertex path. This result is a consequence of a stronger result of Kopylov: if $k$ is odd, $k=2t+1\geq 5$, $n \geq (5t-3)/2$, and $G$ is an $n$-vertex $2$-connected graph with at least $h(n,k,t) := {k-t \choose 2} + t(n -k+ t)$ edges, then $G$ contains a cycle of length at least $k$ unless $G = H_{n,k,t} := K_n - E(K_{n - t})$. In this paper we prove a stability version of the Erd\H{o}s-Gallai Theorem: we show that for all $n \geq 3t > 3$, and $k \in \{2t+1,2t + 2\}$, every $n$-vertex 2-connected graph $G$ with $e(G) > h(n,k,t-1)$ either contains a cycle of length at least $k$ or contains a set of $t$ vertices whose removal gives a star forest. In particular, if $k = 2t + 1 \neq 7$, we show $G \subseteq H_{n,k,t}$. The lower bound $e(G) > h(n,k,t-1)$ in these results is tight and is smaller than Kopylov's bound $h(n,k,t)$ by a term of $n-t-O(1)$.