Local total neighborhood antimagic labelings–a new variant of vertex colorings

Zhen Bin Gao; Gee-Choon Lau; Wai Chee Shiu; Kai Siong Yow; Zhen Bin Gao; Gee-Choon Lau; Wai Chee Shiu; Kai Siong Yow

doi:10.3934/math.2026852

AIMS Mathematics

2026, Volume 11, Issue 7: 20986-21011. doi: 10.3934/math.2026852

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Research article

Local total neighborhood antimagic labelings–a new variant of vertex colorings

1.
College of General Education, Guangdong University of Science and Technology, Dongguan 523083, China
2.
77D, Jalan Suboh, 85000 Segamat, Johor, Malaysia
3.
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China
4.
Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia
5.
Department of Mathematics and Statistics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia

Received: 29 April 2026 Revised: 11 July 2026 Accepted: 13 July 2026 Published: 16 July 2026
MSC : 05C78, 05C15

Let $ G = (V, E) $ be a simple graph of order $ p $ and size $ q $. A local total neighborhood antimagic labeling of $ G $ is a bijection $ f:V\cup E\to[1, p+q] $ such that, for every vertex $ u\in V $, the induced vertex color $ f^+_{tn}(u) = \sum_{ux \in E} [f(ux) + f(x)] $ assigns distinct integers (called colors) to adjacent vertices of $ G $. We define $ f^+_{tn}(u) = 0 $ when $ u $ is an isolated vertex. A graph $ G $ is said to be local total neighborhood antimagic if it admits a local total neighborhood antimagic labeling. The local total neighborhood antimagic chromatic number of $ G $, denoted by $ \chi_{ltna}(G) $, is the minimum number of distinct induced colors taken over all such labelings of $ G $. In this paper, we investigated properties of $ \chi_{ltna}(G) $ for graphs without isolated vertices and provide results that characterize this chromatic parameter. We focused on several graph classes, including paths, cycles, complete graphs, complete bipartite graphs, wheel graphs, and various types of join graphs.
- local total neighborhood antimagic chromatic number,
- path,
- cycle,
- complete graph,
- complete bipartite graph,
- wheel graph,
- join product
Citation: Zhen Bin Gao, Gee-Choon Lau, Wai Chee Shiu, Kai Siong Yow. Local total neighborhood antimagic labelings–a new variant of vertex colorings[J]. AIMS Mathematics, 2026, 11(7): 20986-21011. doi: 10.3934/math.2026852

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Abstract

Let $ G = (V, E) $ be a simple graph of order $ p $ and size $ q $. A local total neighborhood antimagic labeling of $ G $ is a bijection $ f:V\cup E\to[1, p+q] $ such that, for every vertex $ u\in V $, the induced vertex color $ f^+_{tn}(u) = \sum_{ux \in E} [f(ux) + f(x)] $ assigns distinct integers (called colors) to adjacent vertices of $ G $. We define $ f^+_{tn}(u) = 0 $ when $ u $ is an isolated vertex. A graph $ G $ is said to be local total neighborhood antimagic if it admits a local total neighborhood antimagic labeling. The local total neighborhood antimagic chromatic number of $ G $, denoted by $ \chi_{ltna}(G) $, is the minimum number of distinct induced colors taken over all such labelings of $ G $. In this paper, we investigated properties of $ \chi_{ltna}(G) $ for graphs without isolated vertices and provide results that characterize this chromatic parameter. We focused on several graph classes, including paths, cycles, complete graphs, complete bipartite graphs, wheel graphs, and various types of join graphs.

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