Counterexamples to the local antimagic leaf conjecture

Source abstract

It has been conjectured that $χ_{\mathrm{la}}(T)\leq\ell(T)+2$ for every tree $T$ of order at least three, where $χ_{\mathrm{la}}(T)$ is the local antimagic chromatic number and $\ell(T)$ denotes the number of leaves. We disprove this bound and show that no absolute constant can replace $2$ in an inequality of this form. Our approach uses degree-based intervals containing all possible induced vertex weights under bijective edge labelings, together with a separation criterion that forces selected internal vertex weights to be distinct. We first construct caterpillars $T_3$ and $T_4$ satisfying $χ_{\mathrm{la}}(T_3)=\ell(T_3)+3$ and $χ_{\mathrm{la}}(T_4)=\ell(T_4)+4$, respectively. We then construct, for every positive integer $r$, infinitely many pairwise nonisomorphic caterpillars with exactly $r$ internal vertices for which every bijective edge labeling is antimagic. Each such caterpillar $T$ satisfies $χ_{\mathrm{la}}(T)=|V(T)|=\ell(T)+r$. Consequently, $χ_{\mathrm{la}}(T)-\ell(T)$ is unbounded even within the class of caterpillars, and no absolute constant $C$ can satisfy $χ_{\mathrm{la}}(T)\leq\ell(T)+C$ for every tree $T$.