Open research questions in Graph Labeling and Dimension Problems
97 gap statements mined from Graph Labeling and Dimension Problems papers in our 4.5M-paper local library, which holds 732 papers on the topic — drawn mostly from each paper's own stated research gap, future-work, challenge and limitation notes, and its abstract. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one.
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What the literature leaves open
The lack of characterization for s(G□H) for certain values. The need for improvement of the existing upper bound for s(G□H) for some classes of graphs.
The study of graph labeling has emerged as a rich and dynamic area of research, but the exploration of Gaussian integer labellings is relatively new. The paper identifies the need for a novel graph labelling scheme using Gaussian integers applied to Cartesian product graphs.
There is a lack of understanding of the properties of sum graphs over different magmas. There is a lack of methods for finding sum graphs over all magmas.
Further research can be done on the application of the castling procedure to other areas. The study of nil-clean graphs and their automorphism groups can be extended to other rings. The computation of the order of the automorphism groups of nil-clean graphs can be used in various applications.
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.
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 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 study focuses on $\overrightarrow{C_{n}}$-irregular oriented graphs, which may not be generalizable to other types of graphs. The paper does not provide a comprehensive analysis of all possible $\overrightarrow{C_{n}}$-irregular graphs. The study is limited to the construction and analysis of specific graphs.
Future studies can explore the properties and behavior of $\overrightarrow{C_{n}}$-irregular graphs in more detail. The paper suggests investigating the applications of $\overrightarrow{C_{n}}$-irregular graphs in network analysis. Further research can focus on constructing and analyzing $\overrightarrow{C_{n}}$-irregular graphs of higher orders.
The 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 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.
A gap in the understanding of the edit distance function for word-representable and comparability graphs is identified. Prior work has not established bounds for the maximum edit distance of an n-vertex graph from these properties.
The edit distance of word-representable and comparability graphs · 2026Further study of self-identifying codes in other graph families is needed. The application of the results to fault detection in networks should be explored.
The existing literature lacks bounds on the minimum size of self-identifying codes in direct products of complete graphs with paths and cycles. The asymptotic tightness of the bounds is not established.
Future research can focus on the study of Fibonacci numbers of other graph families. The results can be used to study the properties of generalized Fibonacci graphs and their applications in algorithmic questions.
The study of Fibonacci numbers of graphs has been limited to specific graph families, such as paths and trees. There is a need for a more general study of Fibonacci numbers of graphs, focusing on generalized Fibonacci graphs.
There is a gap in our understanding of connected subgraph arrangements, particularly with regard to their factorization. Prior work has not fully characterized the properties of these arrangements.
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 computation of the order of the automorphism groups of nil-clean graphs for the ring ℤn is addressed.
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.
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 paper shows that prior work on semiregular automorphisms is incomplete.
For graphs with the 2-LCA property, we show that this sparse relation, together with the leaf set, determines the full LCA relation after a natural closure operation.
Encoding Phylogenetic Networks with Least Common Ancestor Constraints · 2026To 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.
Most-cited papers in Graph Labeling and Dimension Problems
- Infinite Families of Nontrivial Trivalent Graphs Which are not Tait Colorable · American Mathematical Monthly · 1975 · 344 citations
- On Complementary Graphs · American Mathematical Monthly · 1956 · 314 citations
- On the centrality in a graph · Scandinavian Journal of Psychology · 1974 · 280 citations
- Graph and semigroup homomorphisms on networks of relations · Social Networks · 1983 · 270 citations
- Geographies of Belonging · Environment and Planning A Economy and Space · 2009 · 144 citations
- Total edge irregularity strength of trees · Discussiones Mathematicae Graph Theory · 2006 · 81 citations
- Role colouring a graph · Mathematical Social Sciences · 1991 · 67 citations
- Associative graph products and their independence, domination and coloring numbers · Discussiones Mathematicae Graph Theory · 1996 · 55 citations
- Assignment of Numbers to Vertices · American Mathematical Monthly · 1964 · 55 citations
- Hypergraphs, Entropy, and Inequalities · American Mathematical Monthly · 2004 · 53 citations
Most recent work
- On graphs with equal and different Kromatic symmetric functions · Discrete Applied Mathematics · 2026
- Magic squares on Abelian groups · Discrete Mathematics · 2026
- The Metric Dimension of Sparse Random Graphs · The Electronic Journal of Combinatorics · 2026
- 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
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