8 problems

Let the de Bruijn graph of order $n$ be the directed graph whose vertices are binary words of length $n$, with an edge from $x_1x_2\cdots x_n$ to $x_2\cdots x_ny$ for each…

Non-word-representability conjecture. All simplified de Bruijn graphs $S(n,k)$ are non-word-representable for $n\geq 4$ and $k\geq 3$.

For a positive integer $k$, let $B_k$ be the graph whose vertices are binary strings of length $k$, with an edge between $b_1b_2\ldots b_k$ and $b_2b_3\ldots b_kb_{k+1}$ whenever t…

A binary de Bruijn graph is the de Bruijn graph formed from binary strings, and a necklace is the cycle corresponding to a binary string under cyclic rotation. Golomb's conjecture.…

Let $\mathcal{E}(q,k,\ell)$ denote the maximum number of eBugs in an $\ell$-valid colouring with $q$ colours and $k$ LEDs per eBug. An optimal colouring is one attaining the upper…

Rational generating-function conjecture. If $D$ is a fixed odd prime number, then this exponential generating function is the ratio of two polynomials, each of degree $D-1$. For…

Generation conjecture. Such an analogue would require starting with MISs in $B(d,D)$ for every $d<D$. The conjecture is motivated by the rapid growth in the number of MISs for…

Lichiardopol's conjecture. Equality should hold in both bounds for every odd prime $D$. Equality is known for $D=3$, $5$, and $7$; the assertion for all odd primes is presented as…