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<p>This study introduces an advanced geometric framework for labeling graphs based on visibility from lattice points, establishing two new constructs: visible labeling and weakly visible labeling, which are extensions of coprime and prime labeling, respectively. In this paper, we establish a bijective association between the vertices of a structured graph and integer points on a lattice, with adjacency determined by either a gcd-based lattice visibility condition for visible labeling or a weaker condition parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(b\)</EquationSource> </InlineEquation> for weakly visible labeling. We offer constructive proofs and explicit labeling methods for several graph families, including paths, cycles, complete bipartite graphs, wheel graphs, prism graphs, uniform cycle snake graphs, and binary trees. The framework combines the concepts of coprimality from number theory and visibility from geometry, and we demonstrate that visible labeling is a bounded case of weakly visible labeling, with the density of adjacency increasing monotonically with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(b\)</EquationSource> </InlineEquation>. The two-fold hierarchical framework proposes a graph labeling theory that injects both an algebraic and geometric perspective into structured graph families, resulting in novel higher-dimensional lattice generalizations of visibility concepts and algorithmic graph construction.</p>

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Visible and weakly visible labelings of structured graphs via lattice point visibility

  • 期刊论文

  • 发表日期：

    2026年4月21日

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  • D. Anand,
  • J. Baskar Babujee

摘要

This study introduces an advanced geometric framework for labeling graphs based on visibility from lattice points, establishing two new constructs: visible labeling and weakly visible labeling, which are extensions of coprime and prime labeling, respectively. In this paper, we establish a bijective association between the vertices of a structured graph and integer points on a lattice, with adjacency determined by either a gcd-based lattice visibility condition for visible labeling or a weaker condition parameter \(b\) for weakly visible labeling. We offer constructive proofs and explicit labeling methods for several graph families, including paths, cycles, complete bipartite graphs, wheel graphs, prism graphs, uniform cycle snake graphs, and binary trees. The framework combines the concepts of coprimality from number theory and visibility from geometry, and we demonstrate that visible labeling is a bounded case of weakly visible labeling, with the density of adjacency increasing monotonically with \(b\) . The two-fold hierarchical framework proposes a graph labeling theory that injects both an algebraic and geometric perspective into structured graph families, resulting in novel higher-dimensional lattice generalizations of visibility concepts and algorithmic graph construction.