On Packing Colorings of Distance Graphs
Olivier Togni (Le2i)

TL;DR
This paper investigates the packing chromatic number of infinite distance graphs on integers, providing bounds and specific results for graphs with distances {1,t}, showing finiteness and giving upper bounds for large t.
Contribution
It establishes bounds for the packing chromatic number of infinite distance graphs, especially for graphs with distances {1,t}, including new upper bounds for large t.
Findings
Packing chromatic number is finite for graphs with finite distance sets.
Upper bounds of 40 for odd t and 81 for even t when t ≥ 447.
Main results include bounds for graphs with distances {1,t}.
Abstract
The {\em packing chromatic number} of a graph is the least integer for which there exists a mapping from to such that any two vertices of color are at distance at least . This paper studies the packing chromatic number of infinite distance graphs , i.e. graphs with the set of integers as vertex set, with two distinct vertices being adjacent if and only if . We present lower and upper bounds for , showing that for finite , the packing chromatic number is finite. Our main result concerns distance graphs with for which we prove some upper bounds on their packing chromatic numbers, the smaller ones being for : if is odd and $\chi_{\rho}(G(\mathbb{Z},\{1,t\}))\leq…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
