On a conjecture of Graham and Hنggkvist with the polynomial method

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 v 0 satisfies ρ ( T , v 0 ) ≥ ϕ 1 / 2 , where ϕ = 1 + 5 2 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 Häggkvist and by Lladó and López. The results follow from an application of Alon’s Combinatorial Nullstellensatz to obtain bigraceful labelings.