{
  "id": "math/0512291",
  "version": "v5",
  "published": "2005-12-13T23:00:14.000Z",
  "updated": "2006-01-10T06:37:36.000Z",
  "title": "Some bounds on convex combinations of $ω$ and $χ$ for decompositions into many parts",
  "authors": [
    "Landon Rabern"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A \\emph{$k$--decomposition} of the complete graph $K_n$ is a decomposition of $K_n$ into $k$ spanning subgraphs $G_1,...,G_k$. For a graph parameter $p$, let $p(k;K_n)$ denote the maximum of $\\displaystyle \\sum_{j=1}^{k} p(G_j)$ over all $k$--decompositions of $K_n$. It is known that $\\chi(k;K_n) = omega(k;K_n)$ for $k \\leq 3$ and conjectured that this equality holds for all $k$. In an attempt to get a handle on this, we study convex combinations of $\\omega$ and $\\chi$; namely, the graph parameters $A_r(G) = (1-r) \\omega(G) + r \\chi(G)$ for $0 \\leq r \\leq 1$. It is proven that $A_r(k;K_n) \\leq n + {k \\choose 2}$ for small $r$. In addition, we prove some generalizations of a theorem of Kostochka, et al. \\cite{kostochka}.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2006-01-10T06:37:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "decomposition",
      "graph parameter",
      "study convex combinations",
      "complete graph",
      "equality holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....12291R"
    }
  }
}