Open research questions in Advanced Graph Theory Research
127 gap statements mined from Advanced Graph Theory Research papers in our 4.5M-paper local library, which holds 1,144 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 study of edge pebbling numbers for other families of graphs is still an open research question. The paper identifies a need for further research on the domination cover edge pebbling number for remaining Generalized Petersen graphs and other families of graphs.
On domination cover edge pebbling number of generalized Petersen Graph, Jewel Graph and triangular snake graph · 2026 · DOIThe excerpt is incomplete (Theorem 3.9 proof cuts off mid-sentence at '19'), suggesting missing content that would contain additional results or potential discussion of limitations and future work.
On domination cover edge pebbling number of generalized Petersen Graph, Jewel Graph and triangular snake graph · 2026 · DOIThe lack of a proof for the conjecture that every planar graph without 4-cycles and 5-cycles admits an (F2, Fd)-partition. The need for a new technique to prove the conjecture.
Planar Graphs with Sparse Triangles and Without 4-Cycles and 5-Cycles Admit (\({\mathcal{F}_2, \mathcal{F}_7}\))-Partition · 2026 · DOIThe complexity of the Minimum Stable Cut problem on graphs of low treewidth, low degree, or both was not well understood. There was a lack of efficient algorithms for the problem.
There was an open question about the complexity of computing the degreewidth for oriented graphs. The problem of computing the degreewidth was not well understood.
The lack of understanding of the properties of pt-dominating sets. The lack of knowledge on the relationship among the total domination number, the pt-domination number, and the upper pt-domination number.
The lack of a polynomial characterization of cycle dominating sets. The need for a deeper understanding of the properties of the cycle domination polynomial.
The lack of study on the L-step domination problem. The difficulty in writing out expressions of the L-step domination numbers for an arbitrary list L.
Future research can focus on studying the computational complexity of the dominating covering number. The authors suggest studying the dominating covering number for other graph families.
The gap is the lack of understanding of the dominating covering number and its relation to other graph parameters. The authors identify the need for a comprehensive analysis of the dominating covering number for various graph families.
Isoperimetry was an effective tool when studying the cooling number of grids in, and it may be useful for liminal burning in other graph families. It would be interesting to exchange the goals of the Saboteur and the Arsonist, where the player who reveals vertices wants to minimize the game time, and the player who selects vertices wants to maximize it.
Future research could explore the applications of the paper's results to solving computationally hard problems on graphs of small tree-width. Future research could explore the use of the paper's techniques to capture highly cohesive substructures in other types of graphs.
The paper identifies a need for a more general duality theorem for F-tangles.
The paper addresses the lack of knowledge on the structure of vertex-transitive cubic graphs.
The base size of vertex-transitive cubic graphs · 2026There is a gap in the understanding of the asymptotic values of the minimum number of spanning forests. The paper identifies the need for new bounds and exact values for certain asymptotic values.
On asymptotic values for the minimum number of spanning forests in simple regular graphs · 2026The paper suggests that future research could focus on the applications of the results to other areas of graph theory. The paper proposes that future research could investigate the computational complexity of the algorithms used. The paper recommends that future research could explore the properties of niche graphs and split tournaments in more detail.
The paper notes that prior research has not provided a comprehensive analysis of the characteristics of connected niche graphs of split tournaments. The paper aims to fill this gap.
The domination set construction proofs for coarse deg-centric graphs rely on vertex degree analysis and adjacency counting (as in Propositions 2.17-2.23), but no computational complexity analysis or algorithmic framework for computing the domination number γ(cd(G)) for arbitrary graphs has been provided.
The Propagator gadget (Step 3) enforces consistency across consecutive Selector gadgets but the paper does not explore whether alternative propagation mechanisms or fewer Propagator instances could reduce the overall construction size or improve the hardness lower bound dependence on k and m.
To explore the applications of the degreewidth in different fields.
Prior work did not fully characterize when an independent dominating set is also outer-connected.
The paper identifies a gap in the understanding of domination in coarse deg-centric graphs. The study aims to fill this gap by exploring the fundamental properties and structural characteristics of coarse deg-centric graphs.
The paper identifies a gap in the classification of regular polyhedra. The paper identifies a gap in the understanding of quartic polyhedral Deza graphs.
The study identifies a gap in the understanding of rainbow connectivity in generalized cycle graphs and wheel graphs - Prior work on graph theory and connectivity does not address this specific…
Rainbow Coloring of Cycle of Generalized Cycle and Wheel Graphs for Secured Communication Networks · 2026 · DOIThe burning number conjecture has attracted a great deal of attention in recent years - The paper identifies a gap in the understanding of the burning…
Most-cited papers in Advanced Graph Theory Research
- Optimum Distribution of Switching Centers in a Communication Network and Some Related Graph Theoretic Problems · Operations Research · 1965 · 874 citations
- A Theorem on Graphs, with an Application to a Problem of Traffic Control · American Mathematical Monthly · 1939 · 268 citations
- Algorithm 245: Treesort · Communications of the ACM · 1964 · 200 citations
- A Problem in Graph Theory · American Mathematical Monthly · 1964 · 178 citations
- Worst-Case Analysis of a New Heuristic for the Travelling Salesman Problem · Operations Research Forum · 2022 · 140 citations
- Arrow's theorem with social quasi-orderings · Public Choice · 1984 · 98 citations
- A simple algorithm to detect balance in signed graphs · Mathematical Social Sciences · 1980 · 87 citations
- Single-peaked orders on a tree · Mathematical Social Sciences · 1982 · 76 citations
- A characterization of Roman trees · Discussiones Mathematicae Graph Theory · 2002 · 61 citations
- Graph Theory in Practice: Part I · American Scientist · 2000 · 57 citations
Most recent work
- Structure and linear-Pollyanna for some square-free graphs · Discrete Mathematics · 2026
- Partitioning planar graphs without 4-cycles and 6-cycles into two disjoint subcubic forests · Discrete Mathematics · 2026
- Revisiting d-distance (independent) domination in trees and in bipartite graphs · Discrete Mathematics · 2026
- A remark on a result on odd colorings of planar graphs · Discrete Mathematics · 2026
- Every 4-connected {claw, hourglass}-free graph is 2-Hamiltonian · Discrete Mathematics · 2026
- Perfect divisibility and coloring in fork-free graphs · Discrete Mathematics · 2026
- Oriented trees in digraphs without oriented 4-cycles · Discrete Mathematics · 2026
- Linear recoloring diameter of degenerate chordal graphs and bounded treewidth graphs · Discrete Mathematics · 2026
- Induced subgraphs and tree decompositions XV. Even-hole-free graphs with bounded clique number have logarithmic treewidth. · Transactions of the American Mathematical Society · 2026
- On domination cover edge pebbling number of generalized Petersen Graph, Jewel Graph and triangular snake graph · Journal of Combinatorial Mathematics and Combinatorial Computing · 2026
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