{
  "id": "1401.0836",
  "version": "v1",
  "published": "2014-01-04T18:48:01.000Z",
  "updated": "2014-01-04T18:48:01.000Z",
  "title": "Sequential edge-coloring on the subset of vertices of almost regular graphs",
  "authors": [
    "Petros A. Petrosyan"
  ],
  "comment": "4 pages",
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "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 $n_{r}=\\vert\\{v\\in V(G):d_{G}(v)=r\\}\\vert$. As a corollary, we obtain the following result: if $G$ is a graph with $\\Delta(G)-\\delta(G)\\leq 1$ and $\\chi^{\\prime}(G)=\\Delta(G)=r$ ($r\\geq 3$), then $\\Sigma^{\\prime}(G)\\leq \\left\\lfloor\\frac {2n_{r}(2r-1)+n(r-1)(r^{2}+2r-2)}{4r}\\right\\rfloor$, where $\\Sigma^{\\prime}(G)$ is the edge-chromatic sum of $G$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-01-04T18:48:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "regular graphs",
      "sequential edge-coloring",
      "edges incident",
      "edge-chromatic sum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1401.0836P"
    }
  }
}