Stability in the Erdős–Gallai Theorems on cycles and paths

Dedicated to the memory of G.N. Kopylov
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Abstract

The Erdős–Gallai Theorem states that for k2, every graph of average degree more than k2 contains a k-vertex path. This result is a consequence of a stronger result of Kopylov: if k is odd, k=2t+15, n(5t3)/2, and G is an n-vertex 2-connected graph with at least h(n,k,t):=(kt2)+t(nk+t) edges, then G contains a cycle of length at least k unless G=Hn,k,t:=KnE(Knt).

In this paper we prove a stability version of the Erdős–Gallai Theorem: we show that for all n3t>3, and k{2t+1,2t+2}, every n-vertex 2-connected graph G with e(G)>h(n,k,t1) 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+17, we show GHn,k,t. The lower bound e(G)>h(n,k,t1) in these results is tight and is smaller than Kopylov's bound h(n,k,t) by a term of ntO(1).

Keywords

Turán problem
Cycles
Paths

Cited by (0)

1

Research supported in part by the Hungarian National Science Foundation OTKA 104343, by the Simons Foundation Collaboration Grant 317487, and by the European Research Council Advanced Investigators Grant 267195.

2

Research of this author is supported in part by the National Science Foundation grants DMS-1266016 and DMS-1600592 and by Grant NSh 1939.2014.1 of the President of Russia for Leading Scientific Schools.

3

Research supported by the National Science Foundation Grant DMS-1101489.