An extension of a theorem on cycles containing specified independent edges

  • Authors:

  • Yoshiyas Ishigami;Hong Wang

  • Affiliations:

  • Department of Information and Communication Engineering, The University of Electro-Communications, Tokyo 182-8585, Japan and Department of Mathematics, University of Illinois at Urbana-Champaign, ...;Department of Mathematics, The University of Idaho, Moscow, ID

  • Venue:
  • Discrete Mathematics
  • Year:
  • 2002

Quantified Score

Hi-index 0.05

Visualization

Abstract

We give an alternative proof of a conjecture due to Wang (J. Graph Theory 26 (1997) 105) in a stronger form. The main theorem states that for any integer k ≥ 2 if G is a graph of order n ≥ 4k - 1 and d(u) + d(υ) ≥ n + 2k - 2 for each pair of non-adjacent vertices u and υ of G, then, for any k independent edges e1,... ,ek of G, there exist k vertex-disjoint cycles C1,..., Ck in G such that (i) ei ∈ E(Ci) for all 1 ≤ i ≤ k, (ii) V(C1)∪ ... ∪V(Ck) = V(G), and (iii) #{i ≤ k||Ci| Ka +K2k + Kn-2k-a ⊆ G ⊆ Ka + K2k + Kn-2k-a for some a (2k - 2 a n - 4k + 2). It strengthens the conjecture of Wang, which was first proven by Egawa et al. (Graphs Combin. 16 (2000) 81).