On Graceful Spider Graphs with at Most Four Legs of Lengths Greater than One

Abstract

A graceful labeling of a tree $T$ with $n$ edges is a bijection $f:V(T)\to \{\mathrm{0,1},\mathrm{2},\dots ,n\}$ such that $\{|f(u)-f(v)|:uv\in E(T)\}$ equal to $\{\mathrm{1,2},\dots ,n\}$. A spider graph is a tree with at most one vertex of degree greater than $\mathrm{2}$. We show that all spider graphs with at most four legs of lengths greater than one admit graceful labeling.