Open research questions in Advanced Graph Theory Research
82 unresolved questions extracted from the limitations and future-work sections of 621 Advanced Graph Theory Research papers in our library. Each links back to the study that raised it.
What the literature leaves open
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 gap in the existing literature on tangle-tree duality in infinite graphs. The paper identifies a need for a more general duality theorem for F-tangles.
Future research can study the complexity of list coloring problems on other graph classes. Future research can provide experimental results to validate the algorithm.
The NP-completeness of the problem of determining whether a graph has an interval coloring. The lack of general bounds on W(G(cid:3)H) for all graphs.
The lack of results on the unimodality of the independence polynomial of clique corona graphs. The need to establish several inequalities for the coefficients of the independence polynomial of graphs in Wp.
The Closed Geodetic Game has received less attention than the original Geodetic Game. There is a lack of polynomial-time algorithms for determining Sprague-Grundy values in certain graph classes.
Further study of the properties of the vertex visibility number. Investigation of the vertex visibility number in other graph classes. Analysis of the applications of the vertex visibility number in various domains.
The vertex visibility number is a new concept that has not been thoroughly studied. There is a lack of understanding of the properties of the vertex visibility number. The paper aims to address this gap by investigating the vertex visibility number and its relation to graph structure.
There is a gap in the literature regarding the maximum order of intersecting hypergraphs. The paper aims to fill this gap by systematically studying the maximum order of intersecting hypergraphs.
The complexity of list 3-coloring for comb-convex bipartite graphs was an open problem. The paper resolves this open question.
The existence of Hamilton cycles in connected vertex-transitive graphs is a core open problem in algebraic graph theory, originating from Lovász's 1969 conjecture.
On Hamilton cycles in connected vertex-transitive graphs of order $2pq$ · 2026Finding if EFX alloca- tions always exist, even for agents with additive valuations, is a major open problem in Fair Division.
EFX Allocation In (Multi)Hypergraphs · 2026The problem of finding the isolation number for graphs with larger minimum degree is not well understood. There is a lack of upper bounds on the isolation number for graphs with minimum degree at least four.
This shows the existence of a universal algorithm for the class of all instances in the model with footprints, which is an affirmative answer to the open problem.
Universal Rendezvous of Anonymous Agents with Footprints · 2026This paper aims at answering the open problem from the paper by Das and Pelc (SPAA 2026), asking whether there exists a universal rendezvous algorithm for the class of all instances in the model with footprints.
Universal Rendezvous of Anonymous Agents with Footprints · 2026Beck characterized this challenge in his monograph as the first among the seven most humiliating open problems of positional game theory.
Degree Game for Special Regular Graphs · 2026Breaking this bound for general or even for specific classes of graphs has been a long-standing open problem in combinatorial game theory; indeed, J.
Degree Game for Special Regular Graphs · 2026The gap in prior work is the lack of a satisfactory definition of cohesive subsets. The gap in prior work is the limitation of prior approaches to graphs.
To test LS sets with real data. To identify graph-theoretic properties that are characteristic of subgraphs induced by LS subsets H. To extend these results to hypergraphs.
The need for a best possible bound for diam(G) as a function of n and h. The need to identify graph-theoretic properties that are characteristic of subgraphs induced by LS subsets H.
The nni metric has no known polynomial time algorithm for calculation. The cp distance measure is not well defined.
The 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 paper only examines specific instances of Generalized Petersen graphs (GP6,1, GP6,2, GP7,1) and does not provide a general formula or pattern for computing the domination cover edge pebbling number for arbitrary GPn,k graphs.
On domination cover edge pebbling number of generalized Petersen Graph, Jewel Graph and triangular snake graph · 2026 · DOIThe 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 paper provides domination results for wheel-derived graphs (double wheel, gear, web, sunflower) and cycle-derived graphs (sunlet), but no results are presented for domination in coarse deg-centric graphs of path-based families, tree families, or bipartite graph constructions such as crown graphs or complete bipartite graphs.
Most-cited papers in Advanced Graph Theory Research
- Some results on path-factor critical avoidable graphs · Discussiones Mathematicae Graph Theory · 2020 · 49 citations
- The existence of path-factor covered graphs · Discussiones Mathematicae Graph Theory · 2020 · 22 citations
- The Roman domatic problem in graphs and digraphs: a survey · Discussiones Mathematicae Graph Theory · 2020 · 21 citations
- Coalition graphs of paths, cycles, and trees · Discussiones Mathematicae Graph Theory · 2021 · 16 citations
- Graphs with total mutual-visibility number zero and total mutual-visibility in Cartesian products · Discussiones Mathematicae Graph Theory · 2023 · 16 citations
- Bounds on domination parameters in graphs: a brief survey · Discussiones Mathematicae Graph Theory · 2022 · 12 citations
- Total Roman <i>\{2\}</i> -dominating functions in graphs · Discussiones Mathematicae Graph Theory · 2020 · 12 citations
- Connected coalitions in graphs · Discussiones Mathematicae Graph Theory · 2023 · 11 citations
- Roman <i>\{2\}</i> -domination problem in graphs · Discussiones Mathematicae Graph Theory · 2020 · 7 citations
- The largest Condorcet domain on 8 alternatives · Social Choice and Welfare · 2023 · 6 citations
Most recent work
- Partitioning planar graphs without 4-cycles and 6-cycles into two disjoint subcubic forests · Discrete Mathematics · 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
- On the secure Italian domination in lexicographic product graphs · Asian-European Journal of Mathematics · 2026
- Intersections of Graphs and \({\chi }\)-Boundedness · SIAM Journal on Discrete Mathematics · 2026
- Planar Graphs with Sparse Triangles and Without 4-Cycles and 5-Cycles Admit (\({\mathcal{F}_2, \mathcal{F}_7}\))-Partition · Mathematics · 2026
- Pendant Domination Polynomial of the Corona of a Wheel and an Independent Graph · Annals of Communications in Mathematics · 2026
- A family of transmission irregular trees answering a problem of Stošić and Damnjanović · Computational and Applied Mathematics · 2026
- Secure domination in P5-free graphs · Discrete Mathematics · 2026
- ON DOMINATION IN COARSE DEG-CENTRIC GRAPHS · Advances and Applications in Discrete Mathematics · 2026
- The Leafed Induced Subtree in chordal and bounded treewidth graphs · Discrete Mathematics & Theoretical Computer Science · 2026
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