### The Complexity of 2-Coloring and Strong Coloring in Uniform Hypergraphs with High Degrees

*Edyta Szymanska*

#### Abstract

In this paper we consider the problem of deciding whether a
given $r$-uniform hypergraph $H$ with minimum vertex degree at
least $c\binom{|V(H)|-1}{r-1},$ or minimum degree of a pair of
vertices at least $c\binom{|V(H)|-2}{r-2}$, has a vertex
$2$-coloring. Motivated by an old result of Edwards for graphs,
we obtain first optimal dichotomy results for 2-colorings of
$r$-uniform hypergraphs. For each problem, for every $r\geq 3$ we determine a
threshold value depending on $r$ such that the problem is NP-complete for $c$ below the
threshold, while for $c$ strictly above the threshold it is
polynomial. We provide an algorithm constructing the coloring with
time complexity $O(n^{\lfloor 4/\epsilon\rfloor+2}\log n)$ with some
$\epsilon>0.$ This algorithm becomes more efficient in the case of
$r=3,4,5$ due to known Tur\'{a}n numbers of the triangle and the
Fano plane. In addition, we determine the computational
complexity of strong $k$-coloring of 3-uniform hypergraphs $H$ with
minimum vertex degree at least $c\binom{|V(H)|-1}{2},$ for some
$c,$ leaving a gap for $k\geq 5$ which vanishes as $k\rightarrow \infty$.

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