Open Conjectures in Graph Theory

• Open Conjectures in Graph Theory are unsolved problems probing intrinsic graph properties like cycle structure, coloring, and extremal edge counts that reveal deep combinatorial insights.
• They inspire novel proof techniques and algorithmic strategies, leveraging methods from decomposition analysis to automated conjecture systems in advancing mathematical research.
• These conjectures build interdisciplinary bridges, linking combinatorial, spectral, and algebraic methods to address challenges in discrete mathematics and computational theory.

Open conjectures in graph theory constitute a rich and evolving body of problems that probe the structure, extremal properties, combinatorial invariants, and algorithmic boundaries of graphs. Many of the most significant unsolved questions arise from the paper of cycles, coloring, decomposition, spectral properties, domination, and the interplay between graph structure and number theory. These conjectures not only challenge existing methodologies but also drive the development of new concepts, proof techniques, and computational paradigms.

1. Cycle Structure, Decomposition, and Extremal Edge Problems

Extremal cycle questions remain central to open problems in graph theory. A celebrated topic is the paper of graphs with prescribed restrictions on cycle lengths or patterns and their structural consequences.

• Distinct Cycle Lengths and the Function $f(n)$: The maximum number of edges in an $n$-vertex graph with no two cycles of the same length, defined as

$$f(n) = \max \{ |E(G)| : |V(G)| = n, \text{no two cycles in } G \text{ have the same length} \}.$$

Known bounds include $$f(n) \geq n + \left\lfloor \frac{\sqrt{8n-23}+1}{2} \right\rfloor$$ for $n \geq 3$ and $f(n) < n + 1.98\sqrt{n}$ for large $n$ (1110.1144). It is conjectured that

$$\lim_{n \to \infty} \frac{f(n) - n}{\sqrt{n}} = 12.4.$$

• Turán–Type Extremal Problems for Cycles: For a graph $H$, the Turán number $\operatorname{ex}(n,H)$ is the maximum edge number in an $n$-vertex graph without $H$ as a subgraph. Erdős conjectured that for even cycles,

$\operatorname{ex}(n, C_{2k}) \geq c\, n^{1+1/k}$

for some constant $c>0$ (1110.1144). This question is central to extremal graph theory.

• Cycle Decomposition and the Hajós Conjecture: For simple Eulerian graphs (all degrees even), Hajós conjectured that the minimal number of edge-disjoint cycles covering all edges, or circuit decomposition number $cd(G)$, satisfies

$cd(G) \leq \frac{|V(G)|}{2}.$

While proven for specific classes (e.g., planar, projective, and $K_6$-minor free Eulerian graphs), the general validity remains unresolved (1110.1144).

• Cycle Covering and Related Conjectures for Cubic Graphs: Several conjectures focus on "snarks" (bridgeless cubic graphs that are not 3-edge–colorable), such as the Cycle Double Cover (CDC) conjecture and the Fan–Raspaud conjecture. A structural "complexity scale" has been proposed to measure distance from 3-edge–colorability via perfect matching indices and shortest cycle covers (Máčajová et al., 2020).

2. Coloring, Minors, and Planar Graph Conjectures

Graph coloring and its deep relationship to minors and planarity gives rise to some of the most famous unsolved conjectures.

• Hadwiger's Conjecture: For every integer $t \geq 1$, every $t$-colorable graph has a $K_t$ minor but not a $K_{t+1}$ minor. The conjecture is resolved for $t\leq 6$, but for $t\geq 7$ it remains open (Wang et al., 2021). Novel geometric frameworks, such as “chromatic coordinates” in 3D Euler space (chromatic planes), have been developed to advance the conjecture for $t=8$ (Wang et al., 2021).
• Erdős–Faber–Lovász (EFL) Conjecture: If a graph is the union of $n$ cliques of order $n$, any two of which share at most one vertex, then $\chi(G) = n$. While this is proven for cases where each shared vertex is in exactly two cliques (using explicit coloring algorithms), the general conjecture is open (Gauci et al., 2021).
• 1–2 and 1–2–3 Conjectures: Assigning small integer labels to edges (or vertices and edges) to distinguish adjacent vertices by sums is a major topic. While the 1–2–3 Conjecture is now resolved, variants like the 1–2 Conjecture, particularly for the sum, product, and multiset settings, remain open except for some bounded degree or degeneracy classes (Bensmail et al., 30 Apr 2025).

3. Planar Graphs: Induced Substructures and Domination

A number of central conjectures focus on planar graphs, particularly regarding induced forests and dominating sets.

• Albertson–Berman Conjecture: Every planar graph on $n$ vertices has an induced forest of order at least $n/2$. This is unsolved, despite refinements and breakthroughs on weaker bounds (Enami et al., 12 Jun 2025).
• Matheson–Tarjan Conjecture: Every sufficiently large planar triangulation has a dominating set of size at most $n/4$. Although partial results exist, counterexamples for stronger variants (e.g., for connected domination) demonstrate the subtlety and depth of these problems (Enami et al., 12 Jun 2025).

Induced substructure and treewidth approaches provide new methods, such as the general result that any graph of treewidth $s$ has an induced subgraph of treewidth $t \leq s$ on at least $\frac{t+1}{s+1}|G|$ vertices (Enami et al., 12 Jun 2025).

4. Spectral Graph Theory: Extremal Problems and Automated Refutation

The interface of eigenvalues and graph theory is rich with challenging open problems.

• Spectral Extremal Problems: Open questions often ask for the maximum (or minimum) spectral radius across graphs with forbidden subgraphs, given degree bounds, or planarity restrictions. For instance, the recent resolution that the join $P_2 + P_{n-2}$ uniquely maximizes the spectral radius among planar graphs (Tait et al., 2016) and analogous results for outerplanar and pineapple graphs.
• Nordhaus–Gaddum-type and Q-Spread Problems: Problems such as maximizing $\lambda_1(G) + \lambda_1(\bar{G})$ or the $Q$-spread of the signless Laplacian among connected graphs are archetypal (Liu, 2022).
• Automated Discovery and Refutation: Adaptive Monte Carlo Search (AMCS) and other search-driven or RL algorithms have enabled the automated refutation of conjectures, including the generation of infinite families of counterexamples (Vito et al., 2023, Roucairol et al., 27 Sep 2024, Angileri et al., 18 Jun 2024). Systems now regularly benchmark conjectures, generate candidate inequalities (as in the Optimist–Pessimist dueling framework (Davila, 14 Nov 2024)), and synthesize or rediscover classical results.
• Open Problems in Spectral Graph Theory: Broader surveys present 20+ open topics, from tight spectral radius bounds, signed graph spectra (Bilu–Linial Conjecture), to spectral gaps and the structure of principal eigenvectors (Liu et al., 2023).

5. Metric, Combinatorial, and Game-Theoretic Conjectures

Specialized conjectures connect metric properties, combinatorial game theory, and more abstract notions.

• De Bruijn–Erdős–Type Line Problems: The question whether every finite connected graph of order $n$ has at least $n$ distinct "lines" (in the sense generalized from metric betweenness) or admits a universal line remains open, with lower bounds conjectured for line counts in various families (Chvátal, 2018).
• Graph Decomposition Problems and Synchronization: Conjectures generalizing the road coloring theorem seek minimal synchronizing factors for directed graphs, with bunchy factor conjectures, O(G) conjectures, and their practical verification via computer simulations (Morrison, 2022).
• Game-Theoretic Conjectures: In combinatorial games such as the graph grabbing game, conjectures center on conditions under which the first player can force a win, with subtleties arising from structural properties, forbidden induced subgraphs, and equivalences in binary-weighted cases (Hollom, 2023).

6. Number Theory and Algebraic Connections

Additive combinatorics and number theory inspire new graph constructions and conjectures.

• Goldbach Graphs and Variants: The structure of Goldbach–type graphs, intersections of "prime multiple missing graphs," and their properties (connectivity, diameters, Hamiltonicity) encode unsolved number-theoretic conjectures such as the original Goldbach problem as questions about infinite graphs (Ghosh, 5 Jan 2025).

7. Systematic and Automated Conjecture Generation

Recent advances have produced systems for conjecture formulation, discovery, and testing.

• Automated Conjecture Systems: Programs such as TxGraffiti and the Optimist leverage mixed-integer programming, heuristic filtering, and memory-based computation to propose, refine, and test hundreds of possible graph invariant inequalities. These tools have not only reproduced known theorems but also proposed sharp inequalities awaiting validation or refutation (Caro et al., 2021, Davila, 14 Nov 2024).
• Feedback and Verification Loops: The integration of conjecture-generating and conjecture-challenging agents enables an iterative, fully automated research cycle capable of dynamically updating and refining graph theoretic knowledge (Davila, 14 Nov 2024).

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Open conjectures in graph theory represent the interplay between deep combinatorial structure, algebraic and spectral analysis, number theory, algorithmic challenge, and increasingly, computational and automated approaches. Their resolution, whether by manual insight or automated discovery and refutation, continues to shape the frontiers of both theory and practice in discrete mathematics.