On a conjecture of Graham and Häggkvist with the polynomial method

  • Authors:

  • M. Cámara;A. Lladó;J. Moragas

  • Affiliations:

  • Departament de Matemítica Aplicada IV, Universitat Politècnica de Catalunya, Jordi Girona 1-3, E-08034 Barcelona, Spain;Departament de Matemítica Aplicada IV, Universitat Politècnica de Catalunya, Jordi Girona 1-3, E-08034 Barcelona, Spain;Departament de Matemítica Aplicada IV, Universitat Politècnica de Catalunya, Jordi Girona 1-3, E-08034 Barcelona, Spain

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2009

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Abstract

A conjecture of Graham and Haggkvist states that every tree with m edges decomposes every 2m-regular graph and every bipartite m-regular graph. Let T be a tree with a prime number p of edges. We show that if the growth ratio of T at some vertex v"0 satisfies @r(T,v"0)=@f^1^/^2, where @f=1+52 is the golden ratio, then T decomposes K"2"p","2"p. We also prove that if T has at least p/3 leaves then it decomposes K"2"p","2"p. This improves previous results by Haggkvist and by Llado and Lopez. The results follow from an application of Alon's Combinatorial Nullstellensatz to obtain bigraceful labelings.