Local Antimagic Chromatic Number of Middle Graph and Direct Product of Graphs
Abstract
An edge labeling of a connected graph G=(V, E) is called a local antimagic labeling if it is a bijection f: E→ {1, 2, ⋯, | E | } such that for any pair of adjacent vertices u and v, f+(u)≠ f+(v) where f+(x)=∑ f(e), with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted χla(G), is the minimum number of distinct induced vertex labels across all possible local antimagic labelings of G. This paper investigates two families of graphs G(m[n]) (a graph class constructed by augmenting a bipartite structure B(m[n]) with cyclic edges) and K1,s× Cn (the direct product of a star K1,s and a cycle Cn). Under the conditions odd integer m≥ 1 and even integer n≥ 4, we establish the exact values of χla(G(m[n])) and χla(K1,s× Cn) for various s. Furthermore, we characterize several graph families achieving χla(G)=2 or $3$.
Related
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BibTeX
@misc{xu2026local,
title = {{Local Antimagic Chromatic Number of Middle Graph and Direct Product of Graphs}},
author = {Huimin Xu and Gee-Choon Lau and Ruixue Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14486},
eprint = {2610.14486},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14486},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Local Antimagic Chromatic Number of Middle Graph and Direct Product of Graphs
AU - Huimin Xu
AU - Gee-Choon Lau
AU - Ruixue Zhang
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14486
UR - https://arxiv.org/abs/2610.14486
AB - An edge labeling of a connected graph G=(V, E) is called a local antimagic labeling if it is a bijection f: E→ {1, 2, ⋯, | E | } such that for any pair of adjacent vertices u and v, f^+(u)≠ f^+(v) where f^+(x)=∑ f(e), with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted χ_la(G), is the minimum number of distinct induced vertex labels across all possible local antimagic labelings of G. This paper investigates two families of graphs G(m^[n]) (a graph class constructed by augmenting a bipartite structure B(m^[n]) with cyclic edges) and K_1,s× C_n (the direct product of a star K_1,s and a cycle C_n). Under the conditions odd integer m≥ 1 and even integer n≥ 4, we establish the exact values of χ_la(G(m^[n])) and χ_la(K_1,s× C_n) for various s. Furthermore, we characterize several graph families achieving χ_la(G)=2 or $3$.
ER -