Graph partition into paths containing specified vertices

  • Authors:

  • Ken-ichi Kawarabayashi

  • Affiliations:

  • Department of Mathematics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan and Department of Mathematics, 1326 Stevenson Center, Vanderbilt University, Nashville, TN

  • Venue:
  • Discrete Mathematics
  • Year:
  • 2002

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Abstract

For a graph G, let σ2(G) denote the minimum degree sum of a pair of nonadjacent vertices. Suppose G is a graph of order n. Enomoto and Ota (J. Graph Theory 34 (2000) 163-169) conjectured that, if a partition n = Σi=1k ai is given and σ2(G) Ge; n + k - 1, then for any k distinct vertices υ1,...,υk, G can be decomposed into vertex-disjoint paths P1..., Pk such that |V(Pi)| = ai and υi is an endvertex of Pi. Enomoto and Ota (J. Graph Theory 34 (2000) 163) verified the conjecture for the case where all ai ≤ 5, and the case where k ≤ 3. In this paper, we prove the following theorem, with a stronger assumption of the conjecture. Suppose G is a graph of order n. If a partition n = Σi=1k ai is given and σ2 (G) ≥ Σi=1k max(?? 4/3ai ??, ai + 1) - 1, then for any k distinct vertices υ1,...,υk G can be decomposed into vertex-disjoint paths p1...,Pk such that |V(Pi)| =ai and υi is an endvertex of Pi for all i. This theorem implies that the conjecture is true for the case where all ai ≤ 5 which was proved in (J. Graph Theory 34 (2000) 163-169).