
Efficient Deterministic Distributed Coloring with Small Bandwidth
We show that the (degree+1)list coloring problem can be solved determin...
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Notes on complexity of packing coloring
A packing kcoloring for some integer k of a graph G=(V,E) is a mapping ...
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Colouring Graphs of Bounded Diameter in the Absence of Small Cycles
For k≥ 1, a kcolouring c of G is a mapping from V(G) to {1,2,…,k} such ...
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Distributed Edge Coloring in Time QuasiPolylogarithmic in Delta
The problem of coloring the edges of an nnode graph of maximum degree Δ...
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New DiameterReducing Shortcuts and Directed Hopsets: Breaking the √(n) Barrier
For an nvertex digraph G=(V,E), a shortcut set is a (small) subset of e...
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Coloring Problems on Bipartite Graphs of Small Diameter
We investigate a number of coloring problems restricted to bipartite gra...
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Dynamic Schnyder Woods
A realizer, commonly known as Schnyder woods, of a triangulation is a pa...
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Faster 3coloring of smalldiameter graphs
We study the 3Coloring problem in graphs with small diameter. In 2013, Mertzios and Spirakis showed that for nvertex diameter2 graphs this problem can be solved in subexponential time 2^𝒪(√(n log n)). Whether the problem can be solved in polynomial time remains a wellknown open question in the area of algorithmic graphs theory. In this paper we present an algorithm that solves 3Coloring in nvertex diameter2 graphs in time 2^𝒪(n^1/3log^2 n). This is the first improvement upon the algorithm of Mertzios and Spirakis in the general case, i.e., without putting any further restrictions on the instance graph. In addition to standard branchings and reducing the problem to an instance of 2Sat, the crucial building block of our algorithm is a combinatorial observation about 3colorable diameter2 graphs, which is proven using a probabilistic argument. As a side result, we show that 3Coloring can be solved in time 2^𝒪( (n log n)^2/3) in nvertex diameter3 graphs. We also generalize our algorithms to the problem of finding a list homomorphism from a smalldiameter graph to a cycle.
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