{
  "id": "1506.08345",
  "version": "v1",
  "published": "2015-06-28T02:08:56.000Z",
  "updated": "2015-06-28T02:08:56.000Z",
  "title": "Color-blind index in graphs of very low degree",
  "authors": [
    "Jennifer Diemunsch",
    "Nathan Graber",
    "Lucas Kramer",
    "Victor Larsen",
    "Lauren M. Nelsen",
    "Luke L. Nelsen",
    "Devon Sigler",
    "Derrick Stolee",
    "Charlie Suer"
  ],
  "comment": "10 pages, 3 figures, and a 4 page appendix",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $c:E(G)\\to [k]$ be an edge-coloring of a graph $G$, not necessarily proper. For each vertex $v$, let $\\bar{c}(v)=(a_1,\\ldots,a_k)$, where $a_i$ is the number of edges incident to $v$ with color $i$. Reorder $\\bar{c}(v)$ for every $v$ in $G$ in nonincreasing order to obtain $c^*(v)$, the color-blind partition of $v$. When $c^*$ induces a proper vertex coloring, that is, $c^*(u)\\neq c^*(v)$ for every edge $uv$ in $G$, we say that $c$ is color-blind distinguishing. The minimum $k$ for which there exists a color-blind distinguishing edge coloring $c:E(G)\\to [k]$ is the color-blind index of $G$, denoted $\\operatorname{dal}(G)$. We demonstrate that determining the color-blind index is more subtle than previously thought. In particular, determining if $\\operatorname{dal}(G) \\leq 2$ is NP-complete. We also connect the color-blind index of a regular bipartite graph to 2-colorable regular hypergraphs and characterize when $\\operatorname{dal}(G)$ is finite for a class of 3-regular graphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-28T02:08:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15"
    ],
    "keywords": [
      "color-blind index",
      "low degree",
      "regular bipartite graph",
      "proper vertex",
      "edges incident"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}