Graceful labelings of spiders with three-edge legs and pendant leaves at the center

Abstract

A graph $G$ on $m$ edges is graceful if there is an injection $f : V(G) \to \{0, 1, \ldots, m\}$ whose induced edge labels $\{|f(u) - f(v)| : uv \in E(G)\}$ are exactly $\{1, 2, \ldots, m\}$. Ringel and Kotzig conjectured in 1964 that every tree is graceful. A computer check has confirmed this for all trees on at most 35 vertices (Fang 2010), but no general proof is known. Here we exhibit an infinite family of trees that escapes the named spider results of Bahls--Lake--Wertheim, Panpa--Poomsa-ard, and Panpa--Imnang--Wasuanankul: the family $T(k, m)$ of spiders with $k$ legs of length $3$ together with $m$ pendant leaves at the centre. We prove every such tree is graceful for all $k \ge 1$ and $m \ge 0$. The argument splits into two short lemmas. The first is a pendant-extension lemma that applies whenever the underlying graceful labeling sends the centre to $0$; the second is the base case, namely that $S_{k, 3}$ admits exactly such an apex-zero labeling, a fact already implicit in Bahls--Lake--Wertheim (2010). What is new is the explicit identification of $T(k, m)$ and the observation that the family is closed under both pendant addition and the apex-zero condition, so it includes infinitely many trees not handled by the named theorems above.