{"paper":{"title":"On r-equitable chromatic threshold of Kronecker products of complete graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Wei Wang, Xin Zhang, Zhidan Yan","submitted_at":"2013-10-08T15:57:06Z","abstract_excerpt":"A graph $G$ is $r$-equitably $k$-colorable if its vertex set can be partitioned into $k$ independent sets, any two of which differ in size by at most $r$. The $r$-equitable chromatic threshold of a graph $G$, denoted by $\\chi_{r=}^*(G)$, is the minimum $k$ such that $G$ is $r$-equitably $k'$-colorable for all $k'\\ge k$. Let $G\\times H$ denote the Kronecker product of graphs $G$ and $H$. In this paper, we completely determine the exact value of $\\chi_{r=}^*(K_m\\times K_n)$ for general $m,n$ and $r$. As a consequence, we show that for $r\\ge 2$, if $n\\ge \\frac{1}{r-1}(m+r)(m+2r-1)$ then $K_m\\time"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.2188","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"}