an:05575927
Zbl 1227.05230
Krivelevich, Michael; Sudakov, Benjamin
Minors in expanding graphs
EN
Geom. Funct. Anal. 19, No. 1, 294-331 (2009).
1016-443X 1420-8970
2009
j
05C83 05C35 05C80
extremal problem; graph minors
Summary: We propose a unifying framework for studying extremal problems related to graph minors. This framework relates the existence of a large minor in a given graph to its expansion properties. We then apply the developed framework to several extremal problems and prove in particular that:
{\parindent=7mm
\begin{itemize}\item[(a)]Every \(K_{s,s^\prime}\)-free graph \(G\) with average degree \(r\) (\(2 \leq s \leq s^\prime\) are constants) contains a minor with average degree \(cr^{1+ {\frac{1}{2(s-1)}}}\), for some constant \(c = c(s, s^\prime) > 0\);
\item[(b)]Every \(C_{2k}\)-free graph \(G\) with average degree \(r\) (\(k \geq 2\) is a constant) contains a minor with average degree \(cr^{\frac{k+1}{2}}\), for some constant \(c = c(k) > 0\).
\end{itemize}}
We also derive explicit lower bounds on the minor density in random, pseudo-random and expanding graphs.