On [formula omitted]-labellings of lobsters and trees with a perfect matching

Abstract

A graceful labelling of a tree T is an injective function f:V(T)→{0,…,|E(T)|} such that {|f(u)−f(v)|:uv∈E(T)}={1,…,|E(T)|}. An α-labelling of a tree T is a graceful labelling f with the additional property that there exists an integer k∈{0,…,|E(T)|} such that, for each edge uv∈E(T), either f(u)≤k<f(v) or f(v)≤k<f(u). In this work, we prove that the following families of trees with maximum degree three have α-labellings: lobsters with maximum degree three, without Y-legs and with at most one forbidden ending; trees T with a perfect matching M such that the contraction T∕M has a balanced bipartition and an α-labelling; and trees with a perfect matching such that their contree is a caterpillar with a balanced bipartition. These results are a step towards the conjecture posed by Bermond in 1979 that all lobsters have graceful labellings and also reinforce a conjecture posed by Brankovic, Murch, Pond and Rosa in 2005, which says that every tree with maximum degree three and a perfect matching has an α-labelling.