Bartłomiej Kielak

The $k$-Coloring problem on hereditary graph classes has been a deeply researched problem over the last decade. A hereditary graph class is characterized by a (possibly infinite) list of minimal forbidden induced subgraphs. We say that a graph is $(H_1,H_2,\ldots)$-free if it does not contain any of $H_1,H_2,\ldots$ as induced subgraphs. The complexity landscape of the problem remains unclear even when restricting to the case $k=4$ and classes defined by a few forbidden induced subgraphs. While the case of only one forbidden induced subgraph has been completely resolved lately, the complexity when considering two forbidden induced subgraphs still has a couple of unknown cases. In particular, $4$-Coloring on $(P_6,C_3)$-free graphs is polynomial while it is NP-hard on $(P_{22},C_3)$-free graphs. We provide a reduction showing NP-completeness of $4$-Coloring on $(P_t,C_3)$-free graphs for $19\leq t\leq 21$, thus constricting the gap of cases whose complexity remains unknown. Our proof includes a computer search ensuring that the graph family obtained through the reduction is indeed $P_{19}$-free.

For integers $k, \ell \geq 3$, let $\mathrm{ex}(n, \overrightarrow{C_k}, \overrightarrow{C_\ell})$ denote the maximum number of directed cycles of length $k$ in any oriented graph on $n$ vertices which does not contain a directed cycle of length $\ell$. We establish the order of magnitude of $\mathrm{ex}(n, \overrightarrow{C_k}, \overrightarrow{C_\ell})$ for every $k$ and $\ell$ and determine its value up to a lower error term when $k \nmid \ell$ and $\ell$ is large enough. Additionally, we calculate the value of $\mathrm{ex}(n, \overrightarrow{C_k}, \overrightarrow{C_\ell})$ for some other specific pairs $(k, \ell)$ showing that a diverse class of extremal constructions can appear for small values of $\ell$.

We study the problem of finding the largest number $T(n, m)$ of ternary vectors of length $n$ such that for any three distinct vectors there are at least $m$ coordinates where they pairwise differ. For $m = 1$, this is the classical trifference problem which is wide open. We prove upper and lower bounds on $T(n, m)$ for various ranges of the parameter $m$ and determine the phase transition threshold on $m=m(n)$ where $T(n, m)$ jumps from constant to exponential in $n$. By relating the linear version of this problem to a problem on blocking sets in finite geometry, we give explicit constructions and probabilistic lower bounds. We also compute the exact values of this function and its linear variation for small parameters.