Open research questions in Graph Labeling and Dimension Problems
71 unresolved questions extracted from the limitations and future-work sections of 583 Graph Labeling and Dimension Problems papers in our library. Each links back to the study that raised it.
What the literature leaves open
To explore other families of graphs in relation to the Antimagic Labeling Conjecture. To develop new techniques for constructing partitions of vertices and decompositions of edges. To establish new labeling lemmas that ensure pairwise distinct vertex sums.
The Antimagic Labeling Conjecture remains open for many families of graphs. Bipartite graphs with minimum degree at least 15 have not been fully explored in this context.
To investigate specific applications of Saxl hypergraphs in computer science and engineering. To study the properties of Saxl hypergraphs for specific families of permutation groups. To develop algorithms for computing Saxl hypergraphs.
There is a lack of understanding of the structure of permutation groups with base size greater than 2. The Saxl graph is not applicable to groups with base size greater than 2.
To develop FT labeling schemes for dynamic graphs or graphs with node faults. To improve the tradeoff between space and multiplicative stretch in the routing scheme.
Prior work has only devised compact FT labeling schemes for limited graph families. There is a need for FT connectivity labeling schemes for general graphs.
The conjecture has remained elusive despite decades of effort. The Kempe chain entanglements that defeated Kempe's original proof of the Four Color Theorem pose a challenge. The proof requires a minimal counterexample framework and a Kempe chain contraction technique.
A complete proof of Hadwiger's Conjecture has remained elusive despite decades of effort. Partial results exist for small values of k, but a unified, elementary proof has been missing.
While the chromatic index of the classical helm graph has been determined previously, that for nonuniform pendant extensions remains unexplored.
The Chromatic Index of Asymmetric Pendant Helm Graphs — A Constructive Proof of the Class 1 Property for a Generalized Helm Family · 2026 · DOIFuture research can build on the paper's results to further reduce the complexity of social networks. Future research can explore the application of the paper's results to other domains.
The paper identifies a gap in the existing literature on graph and semigroup homomorphisms. The paper identifies a need to extend the classic approach to blockmodeling via the equivalence of positions.
The paper focuses on specific graph conditions (trees with induced P4, paths with specific leaf structures) but does not address the security number for more general classes of graphs or other product operations beyond Cartesian products.
The paper invokes Vignesh et al. [8] regarding a conjecture on total chromatic numbers but does not discuss whether the 1-factorization methodology presented here resolves related conjectures for line graphs of other graph classes or whether additional structural conditions beyond odd-order constraints are necessary for Type-I classification in broader contexts.
The construction requires determining four distinctly colored vertices (S, T, G, H) and ensuring color parity inheritance for specific edges relative to H and G. However, the paper does not address whether this constraint remains sufficient when extending to line graphs of other dense or symmetric graph families beyond complete graphs, or how the perpendicular axis condition generalizes.
The paper identifies the need to prove that double uniform (t1 l1, t2 l2)-ply is a group A-cordial. The paper identifies the need to decrease the number of cases and avoid the brute force technique.
The authors face the challenge of proving that the mosaic dimension of the matching toggle graph is 5n + 1/2. The paper requires the development of a lattice embedding scheme based on adjacent-edge pairings. The authors need to apply the Sumner-Las Vergnas theorem to guarantee a perfect matching in the line graph.
To study the matching toggle graph of an arbitrary graph. To develop a general formula for the mosaic dimension of the matching toggle graph. To apply the authors' approach to other graph-theoretic problems.
To extend this study to more complex classes of trees. To attempt to establish both conjectures in a more general setting. To prove that every prime graph is also an odd prime graph in general.
The conjecture that every tree is a prime graph remains an interesting open problem. The conjecture that every prime graph also satisfies the odd prime labeling property remains unproven.
Future research can focus on extending the results of the paper to other graph classes. Future research can explore the applications of the paper's findings to various fields, such as computer science and mathematics. Future research can aim to develop new algorithms and models for graph theory based on the paper's results.
The paper identifies a gap in the existing literature, focusing on the preservation of odd prime labeling under duplication operations for three fundamental graph classes. The paper claims that the existing literature does not provide sufficient results for graphs derived from Pn, Cn, and K1,n under various vertex- and edge-duplication constructions.
The problem of finding sets of integers with a maximum number of pairs summing to powers of 2 was not fully solved. The maximum size of graphs of order n that admit such a labeling was not determined for all n ≤ 21.
Maximizing the number of integer pairs summing to powers of 2 via graph labeling and solving restricted systems of linear (in)equations · 2026 · DOIThe total edge irregularity strength of cycle snake graphs has not been studied before. The concept of total edge irregularity strength needs to be extended to cycle snake graphs.
The lack of understanding of group distance magic labelings of cubic graphs. The lack of identification of new families of cubic Γ-distance magic graphs.
Further study of self-identifying codes in other graph families is needed. The application of the results to fault detection in networks should be explored.
Most-cited papers in Graph Labeling and Dimension Problems
- A survey of hereditary properties of graphs · Discussiones Mathematicae Graph Theory · 1997 · 48 citations
- Two sufficient conditions for component factors in graphs · Discussiones Mathematicae Graph Theory · 2021 · 22 citations
- Vertex-antimagic total labelings of graphs · Discussiones Mathematicae Graph Theory · 2003 · 16 citations
- On path-pairability in the Cartesian product of graphs · Discussiones Mathematicae Graph Theory · 2016 · 9 citations
- Antimagic labeling of some biregular bipartite graphs · Discussiones Mathematicae Graph Theory · 2020 · 9 citations
- Vertex partitioning of graphs into odd induced subgraphs · Discussiones Mathematicae Graph Theory · 2020 · 6 citations
- Tree-like isometric subgraphs of hypercubes · Discussiones Mathematicae Graph Theory · 2003 · 6 citations
- Further results on the outer connected geodetic number of a graph · Publications de l Institut Mathematique · 2020 · 5 citations
- Labeling the vertex amalgamation of graphs · Discussiones Mathematicae Graph Theory · 2003 · 5 citations
- On distance magic labelings of Hamming graphs and folded hypercubes · Discussiones Mathematicae Graph Theory · 2021 · 5 citations
Most recent work
- Some new results on the security number in the Cartesian product of graphs · The Art of Discrete and Applied Mathematics · 2026
- Adjacent vertex distinguishing total coloring of new families of graphs · Discrete Mathematics, Algorithms and Applications · 2026
- Exploring Adjacency Recognizable Colorings in Graphs: A New Variant of Adjacency Codes · Contemporary Mathematics · 2026
- On Edouard Product Cordial Labeling of Some Graphs · Annals of Communications in Mathematics · 2026
- On local irregularity conjecture for 2-multigraphs · Applied Mathematics and Computation · 2026
- GAUSSIAN LABELING OF PRODUCT GRAPH IS P · Zenodo (CERN European Organization for Nuclear Research) · 2026
- On forward convex preserving maps between graphs · Discrete Mathematics, Algorithms and Applications · 2026
- ON THE TOTAL CHROMATIC NUMBER OF ODD LINE GRAPHS OF ODD COMPLETE GRAPHS · Advances and Applications in Discrete Mathematics · 2026
- On sum graphs over some magmas · Discrete Mathematics, Algorithms and Applications · 2026
- Central Metric Dimension of the Generalized <i>k</i> –corona G⊙kℋ · Discrete Mathematics, Algorithms and Applications · 2026
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