{"paper":{"title":"The Distance Coloring of Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Lian-Ying Miao, Yi-Zheng Fan","submitted_at":"2012-12-05T14:03:43Z","abstract_excerpt":"Let $G$ be a connected graph with maximum degree $\\Delta \\ge 3$. We investigate the upper bound for the chromatic number $\\chi_\\gamma(G)$ of the power graph $G^\\gamma$. It was proved that $\\chi_\\gamma(G) \\le\\Delta\\frac{(\\Delta-1)^{\\gamma}-1}{\\Delta-2}+1=:M+1$ with equality if and only $G$ is a Moore graph. If $G$ is not a Moore graph, and $G$ holds one of the following conditions: (1) $G$ is non-regular, (2) the girth $g(G) \\le 2\\gamma-1$, (3) $g(G) \\ge 2\\gamma+2$, and the connectivity $\\kappa(G) \\ge 3$ if $\\gamma \\ge 3$, $\\kappa(G) \\ge 4$ but $g(G) >6$ if $\\gamma =2$, (4) $\\Delta$ is sufficie"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.1029","kind":"arxiv","version":3},"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"}