Open research questions in Limits and Structures in Graph Theory
133 gap statements mined from Limits and Structures in Graph Theory papers in our 4.5M-paper local library, which holds 1,000 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.
What the literature leaves open
The previous bound of 3k + 1 for cycles with two blocks was not tight. The paper identifies the need for a tighter bound. The gap is addressed by introducing (k, m)-secant edges and proving a tight bound.
On Subdivision of Cycles with Two Blocks in Chromatic Digraphs Spanned by Hamiltonian Directed Paths · 2026 · DOIThe problem of computing Ramsey numbers is a well-known problem in combinatorics, but only nine nontrivial values of R(r, s) are known. The paper identifies a gap in the literature for book Ramsey numbers.
Future research should investigate the oriented Turán number for other values of k. The study of oriented Turán numbers should be extended to other graphs.
Many intriguing questions regarding the Rank-Ramsey problem remain unanswered. For starters, the growth rate of the function ν d(n) is mostly unknown. For bounded d, we have shown (see Section 4) a polynomial separation between ν d(n) and n, starting at d = 41. Is this true of all d? Do there exist triangle-free Rank-Ramsey graphs? Concretely, Open Problem 8.1. Is there a constant c > 0 such that ν2(n) = O(n1−c)?
Further study of the conditions for a graph to have a 2-factor is needed. The relaxation of the Dirac condition and the Chv´atal-Erd˝os condition can be used to study other graph theory problems.
The authors suggest further research on equitable DP coloring. The paper provides examples and proofs to illustrate the differences between equitable DP coloring and other coloring techniques.
An introduction to equitable DP coloring of graphs · 2026The generation of cubic bipartite graphs is limited to order 42 for girth 6 and order 52 for girth 8. The computer searches are limited to graphs with up to 10 000 vertices.
The conjecture that K3,3, the Heawood graph, and the Pappus graph are the only essentially 4-edge-connected pseudo 2-factor isomorphic cubic bipartite graphs was disproved. There is a need for further study of pseudo 2-factor isomorphic graphs.
The study focuses on random triangle-free graphs. The results are based on the assumption that \"λ ≥ 13 14 log n/n\". The analysis is limited to the case where \"|n -m| ≤ Cn 2/3 \". The study does not consider other types of graphs or structural properties.
Future research can focus on extending the proof to non-planar graphs. Future research can focus on analyzing the properties of k-factor-critical graphs in more detail.
The lack of a proof for planar graphs is a gap in the literature. The conjecture was posed by Favaron and Shi in 1998 but remained unproven for planar graphs.
The paper suggests that further study of the coprime graph and its structural properties is needed. The study of coprime Ramsey numbers could be extended to other variants and generalizations.
The question of whether D/ deg continues to grow at n = 6 is open. The maximum block-sensitivity-to-sensitivity ratio and the maximum decision-tree-to-degree ratio are not well understood for n ≥ 6.
The current research gap is the determination of R¯(F,H) for graphs of size 4. The authors identify the need for a comprehensive analysis of the strong Ramsey numbers for different types of graphs. The paper aims to fill this gap by providing a detailed analysis of the strong Ramsey numbers for graphs of size 4.
The understanding of strong Ramsey numbers is limited. There is a lack of general results for strong Ramsey numbers. The paper identifies a gap in the understanding of strong Ramsey numbers.
The lack of understanding of the chromatic indices for semistrong edge coloring and (0, 1)-relaxed strong edge coloring. The need for a proof of the conjecture about the semistrong chromatic index of connected graphs. The identification of graphs that achieve the bound Δ^2 - 1.
The lack of an upper bound for the strong chromatic index of subquartic graphs. The need for a better understanding of strong edge coloring of graphs.
The construction of matchings M_i via Hall's theorem and Berge's theorem applies to nearly regular graphs, but the extension to graphs with more pronounced degree irregularity (specifically when δ(G) is substantially smaller than ⌊Δ(G)/2⌋) requires new structural conditions beyond those in Theorem 10.
On asymptotically tight bound for the conflict-free chromatic index of nearly regular graphs · 2026 · DOITo extend the method to show improved bounds for all values of d. To apply the results to problems in extremal combinatorics. To further analyze the structure of the hypergraph.
Note on the codegree version of the Erdős--Ko--Rado theorem · 2026The paper suggests that the same method can be used to study Boolean function complexity measures at n = 6, although the enumeration cost will be significantly higher. The results can be used to inform the design of algorithms and circuits for query complexity, communication complexity, and circuit lower bounds.
Investigate the tightness of the upper bound ∆^2 - 1 for all values of ∆. Find infinitely many graphs attaining the bound ∆^2 - 1. Improve the upper bound ∆^2 - 1.
Investigating lower bounds for the conflict-free chromatic index. Exploring applications of the conflict-free chromatic index in graph theory and computer science.
On asymptotically tight bound for the conflict-free chromatic index of nearly regular graphs · 2026 · DOIThe paper identifies a gap in the study of equitable DP coloring. The authors aim to fill this gap by introducing the concept of equitable DP coloring and studying its properties.
An introduction to equitable DP coloring of graphs · 2026Investigating the chromatic number of other types of graphs. Studying the structural properties of random graphs with different edge densities. Examining the relationship between the chromatic number and other graph parameters.
Further study of strong Ramsey numbers for various graphs. Investigation of the strong Ramsey numbers for more general classes of graphs. Exploration of the applications of strong Ramsey numbers.
Most-cited papers in Limits and Structures in Graph Theory
- New methods to color the vertices of a graph · Communications of the ACM · 1979 · 1,113 citations
- Graph colorings with local constraints -- a survey · Discussiones Mathematicae Graph Theory · 1997 · 167 citations
- Three remarks on the many-to-many stable matching problem · Mathematical Social Sciences · 1999 · 131 citations
- On a characterization of stable matchings · Economics Letters · 2000 · 97 citations
- Robbins's Theorem for Mixed Multigraphs · American Mathematical Monthly · 1980 · 97 citations
- AN EXTENSION OF THE GENERALIZED WEBER PROBLEM† · Journal of Regional Science · 1968 · 66 citations
- On k-factor-critical graphs · Discussiones Mathematicae Graph Theory · 1996 · 60 citations
- On a special case of Hadwiger's conjecture · Discussiones Mathematicae Graph Theory · 2003 · 36 citations
- The S-packing chromatic number of a graph · Discussiones Mathematicae Graph Theory · 2012 · 35 citations
- An Electrical Lemma · Mathematics Magazine · 1987 · 35 citations
Most recent work
- Sufficient conditions for a graph with minimum degree to be k-critical with respect to an odd [1,b]-factor · Discrete Mathematics · 2026
- Nonempty interior of pinned distance and tree sets · Advances in Mathematics · 2026
- Finer Control on Relative Sizes of Iterated Sumsets · The Electronic Journal of Combinatorics · 2026
- A Canonical Ramsey Theorem with List Constraints in Random (Hyper-)graphs · The Electronic Journal of Combinatorics · 2026
- On Subdivision of Cycles with Two Blocks in Chromatic Digraphs Spanned by Hamiltonian Directed Paths · Mathematics · 2026
- Random Turán Theorem for Expansions of Spanning Subgraphs of Tight Trees · SIAM Journal on Discrete Mathematics · 2026
- Optimal and Efficient Partite Decompositions of Hypergraphs · 2026
- Gluing posets and the dichotomy of poset saturation numbers · Mathematika · 2026
- Regular subgraphs at every density · Transactions of the American Mathematical Society · 2026
- Grid-based graphs, linear realizations, and the Buratti-Horak-Rosa conjecture · Ars Mathematica Contemporanea · 2026
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