Coloring chains for compression with uncertain priors

Noah Golowich¹

¹Harvard University

The electronic journal of combinatorics, Tome 25 (2018) no. 4

Résumé

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 Erdős et al. We generalize the results of Erdős 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 $\chi(U(N,s,k))$.

Détail

DOI : 10.37236/6468

Classification : 05C15
Mots-clés : chromatic number, nested sequence