Counterexamples to the local antimagic leaf conjecture
Abstract
It has been conjectured that χla(T)≤ℓ(T)+2 for every tree T of order at least three, where χla(T) is the local antimagic chromatic number and ℓ(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 T3 and T4 satisfying χla(T3)=ℓ(T3)+3 and χla(T4)=ℓ(T4)+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 χla(T)=|V(T)|=ℓ(T)+r. Consequently, χla(T)−ℓ(T) is unbounded even within the class of caterpillars, and no absolute constant C can satisfy χla(T)≤ℓ(T)+C for every tree T.
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BibTeX
@misc{brondani2026counterexamples,
title = {{Counterexamples to the local antimagic leaf conjecture}},
author = {André Ebling Brondani and Francisca Andrea Macedo França},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22783},
eprint = {2610.22783},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22783},
}
RIS
TY - UNPB
TI - Counterexamples to the local antimagic leaf conjecture
AU - André Ebling Brondani
AU - Francisca Andrea Macedo França
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.22783
UR - https://arxiv.org/abs/2610.22783
AB - It has been conjectured that χ_la(T)≤ℓ(T)+2 for every tree T of order at least three, where χ_la(T) is the local antimagic chromatic number and ℓ(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_la(T_3)=ℓ(T_3)+3 and χ_la(T_4)=ℓ(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_la(T)=|V(T)|=ℓ(T)+r. Consequently, χ_la(T)−ℓ(T) is unbounded even within the class of caterpillars, and no absolute constant C can satisfy χ_la(T)≤ℓ(T)+C for every tree T.
ER -