A note on short cycles in a hypercube

TL;DR

This paper investigates the maximum edges in hypercube subgraphs without certain cycles, providing bounds and colorings to advance understanding of Erdős's conjecture on quadrilateral-free hypercube subgraphs.

Contribution

It offers a new general upper bound for cycle-free subgraphs in hypercubes when cycle length is of the form 4k+2 and introduces a 4-coloring avoiding monochromatic 10-cycles.

Findings

Established an upper bound for cycle-free subgraphs with cycle length 4k+2.

Constructed a 4-coloring of hypercube edges avoiding monochromatic C_{10}.

Contributed to the understanding of Erdős's conjecture on quadrilateral-free hypercube subgraphs.

Abstract

How many edges can a quadrilateral-free subgraph of a hypercube have? This question was raised by Paul Erd\H{o}s 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 $f(n, C_l) = o(|E(Q_n)|)$, when $l= 4k$, $k\geq 2$ and that $f(n, C_6) \geq \frac{1}{3} |E(Q_n)|$. It is an open question to determine $f(n, C_l)$ for $l=4k+2$, $k\geq 2$. 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 $4$ colors containing no induced monochromatic $C_{10}$.