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- W2906243764 abstract "Let $n, d$ be integers with $1 leq d leq left lfloor frac{n-1}{2} right rfloor$, and set $h(n,d):={n-d choose 2} + d^2$. Erdős proved that when $n geq 6d$, each nonhamiltonian graph $G$ on $n$ vertices with minimum degree $delta(G) geq d$ has at most $h(n,d)$ edges. He also provides a sharpness example $H_{n,d}$ for all such pairs $n,d$. Previously, we showed a stability version of this result: for $n$ large enough, every nonhamiltonian graph $G$ on $n$ vertices with $delta(G) geq d$ and more than $h(n,d+1)$ edges is a subgraph of $H_{n,d}$. In this paper, we show that not only does the graph $H_{n,d}$ maximize the number of edges among nonhamiltonian graphs with $n$ vertices and minimum degree at least $d$, but in fact it maximizes the number of copies of any fixed graph $F$ when $n$ is sufficiently large in comparison with $d$ and $|F|$. We also show a stronger stability theorem, that is, we classify all nonhamiltonian $n$-graphs with $delta(G) geq d$ and more than $h(n,d+2)$ edges. We show this by proving a more general theorem: we describe all such graphs with more than ${n-(d+2) choose k} + (d+2){d+2 choose k-1}$ copies of $K_k$ for any $k$." @default.
- W2906243764 created "2019-01-01" @default.
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- W2906243764 date "2017-03-29" @default.
- W2906243764 modified "2023-09-26" @default.
- W2906243764 title "Extensions of a theorem of ErdH{o}s on nonhamiltonian graphs" @default.
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