{"paper":{"title":"Packing chromatic number versus chromatic and clique number","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bo\\v{s}tjan Bre\\v{s}ar, Douglas F. Rall, Kirsti Wash, Sandi Klav\\v{z}ar","submitted_at":"2017-07-16T16:55:16Z","abstract_excerpt":"The packing chromatic number $\\chi_{\\rho}(G)$ of a graph $G$ is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $V_i$, $i\\in [k]$, where each $V_i$ is an $i$-packing. In this paper, we investigate for a given triple $(a,b,c)$ of positive integers whether there exists a graph $G$ such that $\\omega(G) = a$, $\\chi(G) = b$, and $\\chi_{\\rho}(G) = c$. If so, we say that $(a, b, c)$ is realizable. It is proved that $b=c\\ge 3$ implies $a=b$, and that triples $(2,k,k+1)$ and $(2,k,k+2)$ are not realizable as soon as $k\\ge 4$. Some of the obtained results are deduce"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.04910","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}