{"paper":{"title":"Sequential edge-coloring on the subset of vertices of almost regular graphs","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Petros A. Petrosyan","submitted_at":"2014-01-04T18:48:01Z","abstract_excerpt":"Let $G$ be a graph and $R\\subseteq V(G)$. A proper edge-coloring of a graph $G$ with colors $1,\\ldots,t$ is called an $R$-sequential $t$-coloring if the edges incident to each vertex $v\\in R$ are colored by the colors $1,\\ldots,d_{G}(v)$, where $d_{G}(v)$ is the degree of the vertex $v$ in $G$. In this note, we show that if $G$ is a graph with $\\Delta(G)-\\delta(G)\\leq 1$ and $\\chi^{\\prime}(G)=\\Delta(G)=r$ ($r\\geq 3$), then $G$ has an $R$-sequential $r$-coloring with $\\vert R\\vert \\geq \\left\\lceil\\frac{(r-1)n_{r}+n}{r}\\right\\rceil$, where $n=\\vert V(G)\\vert$ and $n_{r}=\\vert\\{v\\in V(G):d_{G}(v)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.0836","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"}