On a conjecture of Graham and Häggkvist with the polynomial method

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Abstract

A conjecture of Graham and Häggkvist states that every tree with m edges decomposes every 2 m-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 v0satisfies ρ (T, v0) ≥ φ{symbol}1 / 2, where φ{symbol} = frac(1 + sqrt(5), 2) is the golden ratio, then T decomposes K2 p, 2 p. We also prove that if T has at least p / 3 leaves then it decomposes K2 p, 2 p. This improves previous results by Häggkvist and by Lladó and López. The results follow from an application of Alon's Combinatorial Nullstellensatz to obtain bigraceful labelings. © 2009 Elsevier Ltd. All rights reserved.

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Cámara, M., Lladó, A., & Moragas, J. (2009). On a conjecture of Graham and Häggkvist with the polynomial method. European Journal of Combinatorics, 30(7), 1585–1592. https://doi.org/10.1016/j.ejc.2009.03.008

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