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\iteman{io-port 05987727}
\itemau{Dellamonica, Domingos jun.; R\"odl, Vojt\v{e}ch}
\itemti{A note on Thomassen's conjecture.}
\itemso{J. Comb. Theory, Ser. B 101, No. 6, 509-515 (2011).}
\itemab
Summary: In 1983 {\it C. Thomassen} [J. Comb. Theory, Ser. B 35, 129--141 (1983; Zbl 0537.05034)] conjectured that for every $k,g\in N$ there exists $d$ such that any graph with average degree at least $d$ contains a subgraph with average degree at least $k$ and girth at least $g$. A result of {\it L. Pyber}, {\it V. R\"odl} and {\it E. Szemer\'edi} [J. Comb. Theory, Ser. B 63, No.\,1, 41--54 (1995; Zbl 0822.05037)] implies that the conjecture is true for every graph $G$ with average $d(G) \geqslant c_{k,g} \log \Delta (G)$.  We strengthen this and show that the conjecture holds for every graph $G$ with average $D(G) \geqslant \alpha (\log \log \Delta (G))^\beta$ for some constants $\alpha , \beta $ depending on $k$ and $g$.
\itemrv{~}
\itemcc{}
\itemut{girth; Thomassen; regular subgraphs}
\itemli{doi:10.1016/j.jctb.2011.04.002}
\end