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David Mikšaník

Publications and source records attributed to David Mikšaník.

4 recordsLinked to original sources
arXiv2026

On Borodin-Kostochka conjecture for correspondence coloring

Borodin and Kostochka in 1977 conjectured that if a graph $G$ has maximum degree $Δ(G)\ge 9$ and its clique number satisfies $ω(G)\le Δ(G)-1$, then its chromatic number satisfies $χ(G) \le Δ(G)-1$. We prove this statement with respect to a stronger graph coloring parameter, the correspondence chromatic number $χ_{DP}$, provided the maximum degree is sufficiently large. More precisely, we prove that for every integer $Δ\ge 3\cdot 10^9$, a graph $G$ of maximum degree at most $Δ$ satisfies $χ_{DP}(G) \le \max(ω(G),Δ-1)$. This strengthens earlier results of Reed (1999) for usual chromatic number and of Choi, Kierstead and Rabern (2023) for list chromatic number.

math.CO↗
arXiv2026

Unavoidable subgraphs in digraphs with large out-degrees

We ask the question, which oriented trees $T$ must be contained as subgraphs in every finite directed graph of sufficiently large minimum out-degree. We formulate the following simple condition: all vertices in $T$ of in-degree at least $2$ must be on the same 'level' in the natural height function of $T$. We prove this condition to be necessary and conjecture it to be sufficient. In support of our conjecture, we prove it for a fairly general class of trees. An essential tool in the latter proof, and a question interesting in its own right, is finding large subdivided in-stars in a directed graph of large minimum out-degree. We conjecture that any digraph and oriented graph of minimum out-degree at least $k\ell$ and $k\ell/2$, respectively, contains the $(k-1)$-subdivision of the in-star with $\ell$ leaves as a subgraph; this would be tight and generalizes a conjecture of Thomassé. We prove this for digraphs and $k=2$ up to a factor of less than $2$.

math.CO↗
arXiv2024

On Approximately Strategy-Proof Tournament Rules for Collusions of Size at Least Three

A tournament organizer must select one of $n$ possible teams as the winner of a competition after observing all $\binom{n}{2}$ matches between them. The organizer would like to find a tournament rule that simultaneously satisfies the following desiderata. It must be Condorcet-consistent (henceforth, CC), meaning it selects as the winner the unique team that beats all other teams (if one exists). It must also be strongly non-manipulable for groups of size $k$ at probability $α$ (henceforth, k-SNM-$α$), meaning that no subset of $\leq k$ teams can fix the matches among themselves in order to increase the chances any of it's members being selected by more than $α$. Our contributions are threefold. First, wee consider a natural generalization of the Randomized Single Elimination Bracket rule from [Schneider et al. 2017] to $d$-ary trees and provide upper bounds to its manipulability. Then, we propose a novel tournament rule that is CC and 3-SNM-1/2, a strict improvement upon the concurrent work of [Dinev and Weinberg, 2022] who proposed a CC and 3-SNM-31/60 rule. Finally, we initiate the study of reductions among tournament rules.

cs.GT↗