10,528 research outputs found
Distributed Deterministic Edge Coloring using Bounded Neighborhood Independence
We study the {edge-coloring} problem in the message-passing model of
distributed computing. This is one of the most fundamental and well-studied
problems in this area. Currently, the best-known deterministic algorithms for
(2Delta -1)-edge-coloring requires O(Delta) + log-star n time \cite{PR01},
where Delta is the maximum degree of the input graph. Also, recent results of
\cite{BE10} for vertex-coloring imply that one can get an
O(Delta)-edge-coloring in O(Delta^{epsilon} \cdot \log n) time, and an
O(Delta^{1 + epsilon})-edge-coloring in O(log Delta log n) time, for an
arbitrarily small constant epsilon > 0.
In this paper we devise a drastically faster deterministic edge-coloring
algorithm. Specifically, our algorithm computes an O(Delta)-edge-coloring in
O(Delta^{epsilon}) + log-star n time, and an O(Delta^{1 +
epsilon})-edge-coloring in O(log Delta) + log-star n time. This result improves
the previous state-of-the-art {exponentially} in a wide range of Delta,
specifically, for 2^{Omega(\log-star n)} \leq Delta \leq polylog(n). In
addition, for small values of Delta our deterministic algorithm outperforms all
the existing {randomized} algorithms for this problem.
On our way to these results we study the {vertex-coloring} problem on the
family of graphs with bounded {neighborhood independence}. This is a large
family, which strictly includes line graphs of r-hypergraphs for any r = O(1),
and graphs of bounded growth. We devise a very fast deterministic algorithm for
vertex-coloring graphs with bounded neighborhood independence. This algorithm
directly gives rise to our edge-coloring algorithms, which apply to {general}
graphs.
Our main technical contribution is a subroutine that computes an
O(Delta/p)-defective p-vertex coloring of graphs with bounded neighborhood
independence in O(p^2) + \log-star n time, for a parameter p, 1 \leq p \leq
Delta
Facial unique-maximum colorings of plane graphs with restriction on big vertices
A facial unique-maximum coloring of a plane graph is a proper coloring of the
vertices using positive integers such that each face has a unique vertex that
receives the maximum color in that face. Fabrici and G\"{o}ring (2016) proposed
a strengthening of the Four Color Theorem conjecturing that all plane graphs
have a facial unique-maximum coloring using four colors. This conjecture has
been disproven for general plane graphs and it was shown that five colors
suffice. In this paper we show that plane graphs, where vertices of degree at
least four induce a star forest, are facially unique-maximum 4-colorable. This
improves a previous result for subcubic plane graphs by Andova, Lidick\'y,
Lu\v{z}ar, and \v{S}krekovski (2018). We conclude the paper by proposing some
problems.Comment: 8 pages, 5 figure
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