Necessary and sufficient conditions of a class of bipartite graphs with local antimagic chromatic number 2 - an algebraic approach
Abstract
For a connected graph G = (V, E), a bijective edge labeling f:E →{1,… ,|E|} is a local antimagic labeling of G if it induces a vertex labeling f+ such that for any pair of adjacent vertices x and y, f+(x)not= f+(y), where the induced vertex label f+(x)= ∑ f(xu), with u ranging over all the vertices adjacent to x. The minimum number of distinct induced vertex labels over all local antimagic labelings of G is the local antimagic chromatic number of G, denoted χla(G). In this paper, we make use of algebraic analysis to obtain necessary and sufficient conditions for every bipartite graph with all vertices of degree 2 except exactly three vertices of degree at least $3$ (say, u,v,w, such that every (u,v)-, (u,w)- and (v,w)-path is of even length and not all are of length 2) to have local antimagic chromatic number 2. Moreover, we showed that the consecutive edge labels of every induced path of each case is unique.
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BibTeX
@misc{lau2026necessary,
title = {{Necessary and sufficient conditions of a class of bipartite graphs with local antimagic chromatic number 2 - an algebraic approach}},
author = {Gee-Choon Lau and Wai Chee Shiu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.12817},
eprint = {2608.12817},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.12817},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Necessary and sufficient conditions of a class of bipartite graphs with local antimagic chromatic number 2 - an algebraic approach
AU - Gee-Choon Lau
AU - Wai Chee Shiu
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2608.12817
UR - https://arxiv.org/abs/2608.12817
AB - For a connected graph G = (V, E), a bijective edge labeling f:E →{1,… ,|E|} is a local antimagic labeling of G if it induces a vertex labeling f^+ such that for any pair of adjacent vertices x and y, f^+(x)not= f^+(y), where the induced vertex label f^+(x)= ∑ f(xu), with u ranging over all the vertices adjacent to x. The minimum number of distinct induced vertex labels over all local antimagic labelings of G is the local antimagic chromatic number of G, denoted χ_la(G). In this paper, we make use of algebraic analysis to obtain necessary and sufficient conditions for every bipartite graph with all vertices of degree 2 except exactly three vertices of degree at least $3$ (say, u,v,w, such that every (u,v)-, (u,w)- and (v,w)-path is of even length and not all are of length 2) to have local antimagic chromatic number 2. Moreover, we showed that the consecutive edge labels of every induced path of each case is unique.
ER -