Elsevier

Discrete Mathematics

Volume 340, Issue 11, November 2017, Pages 2688-2690
Discrete Mathematics

A stability version for a theorem of Erdős on nonhamiltonian graphs

Dedicated to the memory of Professor H. Sachs
https://doi.org/10.1016/j.disc.2016.08.030Get rights and content
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Abstract

Let n,d be integers with 1dn12, and set h(n,d)nd2+d2 and e(n,d)max{h(n,d),h(n,n12)}. Because h(n,d) is quadratic in d, there exists a d0(n)=(n6)+O(1) such that e(n,1)>e(n,2)>>e(n,d0)=e(n,d0+1)==en,n12.A theorem by Erdős states that for dn12, any n-vertex nonhamiltonian graph G with minimum degree δ(G)d has at most e(n,d) edges, and for d>d0(n) the unique sharpness example is simply the graph KnE(K(n+1)2). Erdős also presented a sharpness example Hn,d for each 1dd0(n).

We show that if d<d0(n) and a 2-connected, nonhamiltonian n-vertex graph G with δ(G)d has more than e(n,d+1) edges, then G is a subgraph of Hn,d. Note that e(n,d)e(n,d+1)=n3d2n2 whenever d<d0(n)1.

Keywords

Turán problem
Hamiltonian cycles
Extremal graph theory

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