Graceful Labeling of Two Families of Spiders

TL;DR

This paper advances the understanding of graceful labelings by establishing new conditions under which certain spider graphs are graceful, and provides explicit labelings for specific spider configurations.

Contribution

It introduces a relaxed condition for constructing graceful graphs from existing ones and proves that certain classes of spiders are always graceful, including explicit labelings for specific cases.

Findings

A new condition for joining a vertex to a path to maintain gracefulness.

Proof that certain spiders with specified leg length conditions are graceful.

Explicit graceful labelings for spiders with one long leg and others of length at most two.

Abstract

A \emph{graceful labeling} of a graph $G$ is an injective function $f : V(G) \to \{0, \ldots, |E(G)|\}$ such that $\{\,|f(u)-f(v)| : uv \in E(G)\,\} = \{1, \ldots, |E(G)|\}$. If such a labeling exists, then we call $G$ \emph{graceful}. Introduced by Rosa in 1967, graceful labeling has been widely studied, and the Graceful Tree Conjecture asserts that every tree is graceful. The conjecture is known to hold for several classes of trees, including caterpillars, trees with at most four leaves, trees of diameter at most five, and certain spiders. An important subclass is that of \emph{$\alpha$-labelings}, where a graceful labeling $f$ admits an integer $\alpha$ such that each edge joins a vertex with label at most $\alpha$ to one with label greater than $\alpha$. A result from 1982 by Huang, Kotzig, and Rosa shows that if $H$ has an $\alpha$-labeling with a vertex $u$ labeled $0$ or $\alpha$, and $G$ has a graceful labeling with a vertex $v$ labeled $0$, then identifying $u$ and $v$ yields a graceful graph, though this requires a $0$-labeled vertex in $G$. We prove a related result that relaxes this condition: if $G$ has a graceful labeling $f$ such that $f(u)+\lfloor n/2 \rfloor + 1 \le n$ and $n \not\equiv 1 \pmod{4}$, where $u\in V(G)$ and $n\ge 2$ is an integer, then joining $u$ to an end vertex of the vertex-disjoint $n$-vertex path $P_n$ yields a graceful graph. As an application, we show that any spider with legs $L_1,\ldots,L_s$ ($s \ge 1$) satisfying $|E(L_{2})| \ge 2|E(L_1)|+ 4$ and $|E(L_{i+1})| \ge 2|E(L_i)|+ 2$ for $i \in \{2,\ldots, s-1\}$ is graceful. Furthermore, we give an explicit graceful labeling for spiders with one leg of arbitrary length and all others of length at most two such that the center is labeled by $0$. This labeling enables the construction of larger graceful spiders by attaching paths at the center.