Coloring Chains for Compression with Uncertain Priors

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 $U(N,s,k)$ for parameters $N,s,k$, 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 $U(N,s,k)$ 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 $H$ and $G$ when there is a homomorphism $\phi :H\rightarrow G$ that satisfies a nice property. We then use these results to improve upper and lower bounds on $\chi(U(N,s,k))$.