Graph Coloring and Irregularity Strength Analysis

Graph colouring and irregularity strength represent two complementary strands of graph-labelling theory with deep roots in combinatorial optimisation and practical implications for network design, scheduling and resource allocation. At its core, graph colouring assigns labels (commonly called colours) to vertices or edges so that adjacent elements receive distinct labels, yielding proper vertex- or edge-colourings. This framework extends naturally to weighted colourings in which integer weights on edges induce vertex colours via summation, leading to neighbour sum-distinguishing and related indices.

Irregularity strength focuses on edge labellings that guarantee distinct edge-weight sums when each edge weight is combined with the labels of its end vertices. The principal invariant, the irregularity strength of a graph, is the smallest integer k for which such an assignment exists. Variants include total irregularity strength, reflexive edge strength and modular irregularity strength, each refining the labelling rules or arithmetic domain. Applications range from frequency assignment in wireless networks to symmetry breaking in chemical and communication graphs. Recent advances have leveraged algorithmic enumeration, structural decompositions into locally regular and irregular components, and novel algebraic labelling schemes to tighten bounds and explore new graph families.

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Research from all publishers

A study of wheel-related graphs has introduced edge odd graceful labelling for several new families, characterising necessary and sufficient conditions under which these graphs admit labellings that assign odd integers to edges and yield distinct induced vertex sums. This work expands the catalogue of graphs known to satisfy odd graceful criteria and suggests pathways for extending graceful-labelling conjectures.

A 2023 investigation into modular edge irregularity strength defined a vertex k-labelling whose induced edge weights are taken modulo the total number of edges. Sharp lower and upper bounds were established, and precise strength values were determined for classic graphs such as caterpillars, cycles, friendship graphs and n-sun configurations, demonstrating the tightness of these bounds and illustrating the interplay between global graph structure and modular arithmetic constraints.

An algorithmic approach to circulant graphs has yielded new upper bounds on irregularity strength by decomposing complete graphs into circulant subgraphs. Iterative computational labelling algorithms provide explicit k-values for large parameter ranges and suggest asymptotic bounds of order ∣E∣/(2 log2 ∣V∣). This work highlights the power of computer-assisted methods in tackling challenging labelling problems for highly symmetric graphs.

Graph Coloring and Irregularity Strength Analysis publication trend

The graph below shows the total number of articles in graph coloring and irregularity strength analysis across all publications each year (not limited to Nature Index journals).

Technical terms

Graph colouring: Assignment of labels (colours) to vertices or edges so that adjacent elements receive distinct labels.

Edge labelling: Mapping from the edge set of a graph to a set of integers, often subject to distinctness or sum-based constraints.

Irregularity strength: The minimum integer k such that edges and vertices can be labelled from {1,…,k} (or with related constraints) so that each edge’s combined weight is unique.

Total irregularity strength: A variant where both vertices and edges receive labels and the irregularity criterion applies to edge-weight sums including incident vertex labels.

Modular edge irregularity strength: The least k for which vertex labelling yields edge weights taken modulo ∣E∣ that are pairwise distinct.

References

1. Irregularity Strength of Circulant Graphs Using Algorithmic Approach. IEEE Access (2021).
2. Edge Odd Graceful Labeling in Some Wheel-Related Graphs. Mathematics (2024).
3. Neighbor Sum Distinguishing Index. Graphs and Combinatorics (2012).
4. Modular edge irregularity strength of graphs. AIMS Mathematics (2023).
5. Edge-partitioning graphs into regular and locally irregular components. Discrete Mathematics & Theoretical Computer Science (2016).