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- W2949449304 abstract "A vertex-coloring of a hypergraph is conflict-free, if each edge contains a vertex whose color is not repeated on any other vertex of that edge. Let $f(r, Delta)$ be the smallest integer $k$ such that each $r$-uniform hypergraph of maximum vertex degree $Delta$ has a conflict-free coloring with at most $k$ colors. As shown by Tardos and Pach, similarly to a classical Brooks' type theorem for hypergraphs, $f(r, Delta)leq Delta+1$. Compared to Brooks' theorem, according to which there is only a couple of graphs/hypergraphs that attain the $Delta+1$ bound, we show that there are several infinite classes of uniform hypergraphs for which the upper bound is attained. We provide bounds on $f(r, Delta)$ in terms of~$Delta$ for large~$Delta$ and establish the connection between conflict-free colorings and so-called ${t, r-t}$-factors in $r$-regular graphs. Here, a ${t, r-t}$-factor is a factor in which each degree is either $t$ or $r-t$. Among others, we disprove a conjecture of Akbari and Kano~[Graphs and Combinatorics 30(4):821--826, 2014] stating that there is a ${t,r-t}$-factor in every $r$-regular graph for odd $r$ and any odd $t<frac{r}{3}$." @default.
- W2949449304 created "2019-06-27" @default.
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- W2949449304 date "2014-10-05" @default.
- W2949449304 modified "2023-09-27" @default.
- W2949449304 title "Brooks Type Results for Conflict-Free Colorings and {a, b}-factors in graphs" @default.
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