A randomized one-pass insertion-only algorithm that colors an $n$-vertex $k$-uniform hypergraph of maximum degree $\Delta$ with $O(\Delta^{1/(k-1)})$ colors using $\widetilde O_k(n)$ bits of working memory is given.
Abstract
For every fixed $k\ge2$, we give a randomized one-pass insertion-only algorithm that colors an $n$-vertex $k$-uniform hypergraph of maximum degree $\Delta$ with $O(\Delta^{1/(k-1)})$ colors using $\widetilde O_k(n)$ bits of working memory. As a graph-theoretic result of independent interest, we also prove a tight palette-sparsification theorem for general uniform hypergraphs. Independently sampled lists of $\Theta(\sqrt{\log n})$ colors from a palette of size $O(\Delta^{1/(k-1)})$ preserve colorability with high probability; the list-size dependence is asymptotically optimal. These results extend to bounded-rank hypergraphs. We complement the algorithm with a deterministic lower bound: for every fixed polylogarithmic semi-streaming space bound, there are polylogarithmic values of $\Delta$ for which any deterministic one-pass algorithm requires $\exp(\Delta^{\Omega(1)})$ colors.
We study rainbow spanning configurations in bijumbled graphs whose edges are coloured independently and uniformly from a prescribed palette. For $n$-vertex $(p,\beta)$-bijumbled graphs with minimum degree at least a fixed positive multiple of $pn$, we obtain rainbow perfect matchings and Hamilton cycles with a sufficie...
Elad Aigner-Horev, Dan Hefetz, Yury Person et al.· 1 citation
We study the largest size $C(n)$ of a subset of the $n\times n$ integer grid containing no isosceles triple, including equally spaced collinear triples. We prove $C(n)=\Omega\left(n\sqrt{\log\log n/\log n}\right)$ by applying the sparse hypergraph coloring theorem of Cooper and Mubayi. Elementary geometry and primitive...
Given any fixed integer $k \ge 2$ and sufficiently large $d$, we show that the largest possible fractional chromatic number of a $k$-uniform $d$-degenerate uncrowded hypergraph $H$ (i.e., with girth at least $5$) satisfies \[ \chi_f(H) = (1 + o_d(1)) \left((k-1)\,\frac{d}{\log d}\right)^{\frac{1}{k-1}}. \] In fact, we...
Abhishek Dhawan, Li-Na Li, Abhishek Methuku et al.· 0 citations
The first $poly(\Delta,\log n)-round algorithm for $(\Delta + 1)$-edge coloring in the CONGEST model is presented and the $n$-dependency of its runtime, $\tilde{O}(\log^5 n)$, matches the best published dependency in the LOCAL model.
Sebastian Brandt, Ananth Narayanan, Alexandre Nolin· 0 citations
We prove a sparse version of the four-colour theorem of Brandt and Thomass\'{e}, answering a question of Allen, B\"ottcher, Kohayakawa and Roberts. For every fixed $0<\gamma\le1/10$ and every $p=p(n)\in(0,1]$, asymptotically almost surely every spanning triangle-free $H\subseteq G(n,p)$ with $\delta(H)\ge(1/3+\gamma)pn...
We give a randomized polynomial-time algorithm that, for every fixed $\varepsilon>0$, colors every $3$-colorable $n$-vertex graph using $O\bigl(n^{(13-\sqrt{97})/18+\varepsilon}\bigr) \approx O\bigl(n^{0.17506+\varepsilon}\bigr)$ colors, improving upon the previous best bound of $O(n^{0.19539})$ from Bansal, Huang, and...
Ijay Narang, Yu-Kai Tang· 1 citation· ⚡1
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
