A note on short cycles in a hypercube

Abstract

How many edges can a quadrilateral-free subgraph of a hypercube have? This question was raised by Paul Erdős about 27 years ago. His conjecture that such a subgraph asymptotically has at most half the edges of a hypercube is still unresolved. Let $f(n,C_l)$ be the largest number of edges in a subgraph of a hypercube $Q_n$ containing no cycle of length l. It is known that $f(n,C_l)=\mathrm{o}(|E(Q_n)|)$, when $l=4k$, $k⩾2$ and that $f(n,C_6)⩾\frac{1}{3}|E(Q_n)|$. It is an open question to determine $f(n,C_l)$ for $l=4k+2$, $k⩾2$. Here, we give a general upper bound for $f(n,C_l)$ when $l=4k+2$ and provide a coloring of $E(Q_n)$ by four colors containing no induced monochromatic $C_{10}$.