On α-labellings of lobsters and trees with a perfect matching

A graceful labelling of a tree T is an injective function f: V(T) -> {0, ..., vertical bar E(T)vertical bar} such that {vertical bar f(u) - f (v)vertical bar: uv is an element of E(T)} = {1, ..., vertical bar E(T)vertical bar}. An alpha-labelling of a tree T is a graceful labelling f with the additional property that there exists an integer k is an element of {0, ..., vertical bar E(T)vertical bar} such that, for each edge uv is an element of 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 alpha-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 alpha-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 alpha-labelling. (C) 2019 Elsevier B.V. All rights reserved.