Every tree is a large subtree of a tree that decomposes Kn or Kn,n

• Published in: Discrete mathematics Vol. 310; no. 4; pp. 838 - 842
• Main Authors: LLADO, A, LOPEZ, S. C, MORAGAS, J
• Format: Journal Article Publication
• Language: English
• Published: Kidlington Elsevier 28.02.2010
• Subjects: Algebra; Combinatorics; Combinatorics. Ordered structures; Exact sciences and technology; Grafs, Teoria de; Graph labelings; Graph theory; Matemàtica discreta; Matemàtiques i estadística; Mathematics; Sciences and techniques of general use; Teoria de grafs; Àrees temàtiques de la UPC; Integer; Graph labelings; Graph decomposition; Graph decompositions; Complete graph; Discrete mathematics; Tree; Bipartite graph; Combinatorics; Graph labelling; Combinatorial nullstellensatz
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2009.09.021

Let T be a tree with m edges. A well-known conjecture of Ringel states that T decomposes the complete graph $K_{2m+1}$. Graham and Häggkvist conjectured that T also decomposes the complete bipartite graph $K_{m,m}$. In this paper we show that there exists an integer n with n ≤[(3m - 1)/2] and a tree T₁ with n edges such that T₁ decomposes $K_{2n+1}$ and contains T. We also show that there exists an integer n' with n' ≥ 2m-1 and a tree T₂ with n' edges such that T₂ decomposes $K_{n',n'}$and contains T. In the latter case, we can improve the bound if there exists a prime p such that [3m/2] ≤ p < 2m - 1.