Coloring Chains for Compression with Uncertain Priors
Noah Golowich

TL;DR
This paper advances the understanding of graph coloring bounds related to compression schemes with uncertain priors, providing new theoretical bounds and generalizations for specific graph classes.
Contribution
It generalizes previous results by establishing bounds on chromatic numbers of graphs connected via homomorphisms, improving bounds for graphs relevant to compression with uncertain priors.
Findings
Derived new bounds on chromatic numbers of graphs with homomorphisms.
Extended results of Erdős et al. to broader classes of graphs.
Improved upper and lower bounds on $U(N,s,k)$.
Abstract
Haramaty and Sudan considered the problem of transmitting a message between two people, Alice and Bob, when Alice's and Bob's priors on the message are allowed to differ by at most a given factor. To find a deterministic compression scheme for this problem, they showed that it is sufficient to obtain an upper bound on the chromatic number of a graph, denoted for parameters , whose vertices are nested sequences of subsets and whose edges are between vertices that have similar sequences of sets. In turn, there is a close relationship between the problem of determining the chromatic number of and a local graph coloring problem considered by Erd\H{o}s et al. We generalize the results of Erd\H{o}s et al. by finding bounds on the chromatic numbers of graphs and when there is a homomorphism that satisfies a nice property. We then use…
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