Counterexamples to the local antimagic leaf conjecture

It has been conjectured that ${\chi }_{\mathrm{l}\mathrm{a}}(T)\le \mathrm{\ell }(T)+2$ for every tree $T$ of order at least three, where ${\chi }_{\mathrm{l}\mathrm{a}}(T)$ is the local antimagic chromatic number and $\mathrm{\ell }(T)$ 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 ${\chi }_{\mathrm{l}\mathrm{a}}(T_3)=\mathrm{\ell }(T_3)+3$ and ${\chi }_{\mathrm{l}\mathrm{a}}(T_4)=\mathrm{\ell }(T_4)+4$, 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 ${\chi }_{\mathrm{l}\mathrm{a}}(T)=\mathrm{\mid }V(T)\mathrm{\mid }=\mathrm{\ell }(T)+r$. Consequently, ${\chi }_{\mathrm{l}\mathrm{a}}(T)-\mathrm{\ell }(T)$ is unbounded even within the class of caterpillars, and no absolute constant $C$ can satisfy ${\chi }_{\mathrm{l}\mathrm{a}}(T)\le \mathrm{\ell }(T)+C$ for every tree $T$.