Open research questions in Limits and Structures in Graph Theory

What the literature leaves open

• 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.

  New type degree conditions for a graph to have a 2-factor · 2026 · DOI

• The characterization of graphs for which μ(G) = 1 and μ(G) = 2 was not known. The paper identifies a gap in the understanding of the connection between edge-coloured graphs and quantum physics.

  Edge-Coloured Graphs with only Monochromatic Perfect Matchings and Their Connection to Quantum Physics · 2026 · DOI

• The lack of knowledge on the Alon-Tarsi number for K_{3,3}-minor-free graphs. The need for a tight upper bound on the Alon-Tarsi number for K_{s,t}-minor-free graphs.

  The Alon-Tarsi number of $K_{3,3}$-minor-free graphs · 2026 · DOI

• To further improve the constant in the Crossing Lemma. To apply the new technique to other graph classes. To explore the applications of the paper's results.

  Improving the Crossing Lemma by Characterizing Dense 2-Planar and 3-Planar Graphs · 2026 · DOI

• The paper identifies a gap in the current bounds for the crossing number of a given graph. The paper identifies a need for a new technique to characterize dense 2-planar and 3-planar graphs.

  Improving the Crossing Lemma by Characterizing Dense 2-Planar and 3-Planar Graphs · 2026 · DOI

• Future research should aim to generalize the results to all cycles in random graphs. Future research should aim to apply the results to other areas of study, such as social networks and complex systems.

  A canonical Ramsey theorem for even cycles in random graphs · 2026 · DOI

• There is a gap in the current understanding of the thresholds for certain properties in random graphs. The current results do not provide a general result for all cycles in random graphs.

  A canonical Ramsey theorem for even cycles in random graphs · 2026 · DOI

• The lack of a solution for the specific case of 6-cycles is a gap in the existing literature. The problem of maximizing the number of colour alternating cycles in edge-coloured graphs is not fully solved.

  On Alternating 6-Cycles in Edge-Coloured Graphs · 2026 · DOI

• Determining exact values or improving bounds for famous partial cubes. Applying the results to other areas where pursuit-evasion models are relevant. Exploring the properties of other classes of graphs related to partial cubes.

  Cop Number of Partial Cubes · 2026 · DOI

• Little is known about the cop number of partial cubes. The existing bounds for median graphs may not be tight. There is a gap between the lower and upper bounds for the cop number of partial cubes.

  Cop Number of Partial Cubes · 2026 · DOI

• The lack of a direct connection between the chromatic number and the incidence-free number. The lack of a construction of an incidence-free pair for PG(2,q^2).

  A note on the chromatic number of Kneser graphs on chambers of projective planes and incidence-free sets · 2026 · DOI

• Future research could explore further generalizations of the Kruskal-Katona theorem and the Friedgut-Kahn theorem. Future research could explore applications of the paper's results to other areas of mathematics.

  When Joints Meet Extremal Graph Theory: Hypergraph Joints · 2026 · DOI

• The partial shadow problem was not previously generalized to any hypergraph. The Kruskal-Katona theorem and the Friedgut-Kahn theorem were not previously generalized in this way.

  When Joints Meet Extremal Graph Theory: Hypergraph Joints · 2026 · DOI

• To extend the results to other types of graphs. To improve the bounds for the maximal number of edges. To apply the results to practical problems.

  The edge-isoperimetric inequality for powers of cycles · 2026 · DOI

• The edge-isoperimetric problem for powers of cycle graphs was not previously solved. There was a need for a new approach to solve the problem.

  The edge-isoperimetric inequality for powers of cycles · 2026 · DOI

• The previous result by Chen and Deng does not consider odd minor versions. The gap is addressed by considering odd H-models.

  Odd complete bipartite minors in graphs with independence number two · 2026 · DOI

• The problem of finding conditions for a graph to have a 2-factor is not well understood. Prior work has focused on Hamilton cycles, but 2-factors are also of interest.

  New type degree conditions for a graph to have a 2-factor · 2026 · DOI

• We determine the sharp constant in an open problem of Nikiforov (2008) on cycles of consecutive lengths.

  Nikiforov's spectral consecutive cycle problem and the connected-matching method · 2026

• We resolve this long-standing open problem by generalizing and combining tools from the $(k+2)$-coloring to $k$-list-coloring reduction of [Zamir, ICALP 2021] and the hypergraph-containers based approach in [Zamir, STOC 2023].

  k-Coloring is Faster than Computing the Chromatic Number · 2026

• A fundamental open question about DP color functions asks whether, for every graph $G$, there exist $N \in \mathbb{N}$ and a polynomial $p$ such that $P_{DP}(G,q) = p(q)$ whenever $q \geq N$.

  The DP Color Function of Bipartite Graphs · 2026

• While the chromatic thresholds have been completely determined, rather surprisingly the structural behaviors of extremal graphs near the threshold remain unexplored.

  Stability with Minuscule Structure for Chromatic Thresholds · 2026 · DOI

• 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 · DOI

• The random graph model analysis via Observation 11 establishes that χ'_CF(G) = (1 + o(1)) log₂ Δ a.a.s. for G(n,p) with p ≫ n^(-ε), but the behavior for sparse random graphs with p ≤ n^(-ε) and intermediate density regimes is not addressed.

  On asymptotically tight bound for the conflict-free chromatic index of nearly regular graphs · 2026 · DOI

• The 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.

  Lower bounds for book Ramsey numbers · 2026 · DOI

• 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.

  Extremal oriented graphs avoiding 1-subdivision of an in-star · 2026 · DOI

Most-cited papers in Limits and Structures in Graph Theory

• A survey on packing colorings · Discussiones Mathematicae Graph Theory · 2020 · 26 citations
• Bipartite exponential random graph models with nodal random effects · Social Networks · 2021 · 11 citations
• Generalized Turán problems for small graphs · Discussiones Mathematicae Graph Theory · 2021 · 9 citations
• Extremal digraphs avoiding distinct walks of length 4 with the same endpoints · Discussiones Mathematicae Graph Theory · 2020 · 8 citations
• More on the rainbow disconnection in graphs · Discussiones Mathematicae Graph Theory · 2020 · 8 citations
• General sharp upper bounds on the total coalition number · Discussiones Mathematicae Graph Theory · 2023 · 7 citations
• The Turán number of three disjoint paths · Discussiones Mathematicae Graph Theory · 2023 · 6 citations
• Independence number and packing coloring of generalized Mycielski graphs · Discussiones Mathematicae Graph Theory · 2020 · 6 citations
• Graphs with 4-rainbow index 3 and n-1 · Discussiones Mathematicae Graph Theory · 2015 · 5 citations
• Coloring squares of planar graphs with small maximum degree · Discussiones Mathematicae Graph Theory · 2022 · 4 citations

Most recent work

• 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
• Ramsey numbers of trees versus generalized wheels · Discrete Mathematics · 2026
• On asymptotically tight bound for the conflict-free chromatic index of nearly regular graphs · Discrete Mathematics · 2026
• Sufficient conditions for edge-colored bipartite graphs to have rainbow and properly colored spanning trees · Discrete Mathematics · 2026
• Odd cycle-nice claw-free graphs · Indian Journal of Pure and Applied Mathematics · 2026
• Lower bounds for book Ramsey numbers · Discrete Mathematics · 2026
• Extremal oriented graphs avoiding 1-subdivision of an in-star · Discrete Applied Mathematics · 2026
• The Rank-Ramsey problem and the Log-Rank conjecture · Combinatorics, Probability and Computing · 2026

Related topics in Mathematics