Unsolved problems in graph theory

Graph theory is a rich area of mathematics with many interesting unsolved problems. Here are some notable ones: 1. **Graph Isomorphism Problem**: This problem asks whether two finite graphs are isomorphic, meaning they have the same structure regardless of the labels of their vertices. While there are polynomial-time algorithms for certain classes of graphs, a general polynomial-time solution for all graphs remains elusive.

"Erdős on Graphs" typically refers to the collection of works and contributions made by the famous Hungarian mathematician Paul Erdős in the field of graph theory. Erdős is known for his prolific output and collaborations, having published thousands of papers, many of which have shaped the development of various areas in mathematics, including combinatorics and graph theory.

The Erdős–Gyárfás conjecture is a statement in the field of graph theory that pertains to the coloring of graphs. Specifically, it suggests that for any graph \( G \) that does not contain a complete bipartite subgraph \( K_{p,q} \) (i.e.

Harborth's conjecture is a hypothesis in the field of graph theory, particularly related to the properties of planar graphs. Specifically, it suggests that every planar graph can be colored using at most four colors such that no adjacent vertices share the same color. This assertion is closely related to the well-known Four Color Theorem, which states that four colors are sufficient to color the vertices of any planar graph.

The Lovász–Woodall conjecture is a conjecture in graph theory related to the concept of an ideal vertex cover and the independence number of graphs. Specifically, it provides a bound on the size of a minimum dominating set in terms of the independence number and the number of edges in a graph.

The Oberwolfach problem is a problem in combinatorial design and graph theory that involves the arrangement of pairs (or "couples") of items, typically represented as graphs or edges. It is named after the Oberwolfach Institute for Mathematics in Germany, where the problem was first studied. The classical statement of the problem can be described as follows: You have a finite group of \( n \) people (or vertices) who need to meet in pairs over a series of days (or rounds).

The Reconstruction Conjecture is a concept in the field of graph theory, specifically related to the properties of graphs. It posits that a simple graph (i.e., a graph without loops or multiple edges) can be uniquely determined (reconstructed) from the collection of its vertex-deleted subgraphs.

The Second Neighborhood Problem is a concept in the field of graph theory and network analysis, particularly relevant in the study of social networks and community detection. It is often associated with the analysis of local structures within a network. In this context, the "first neighborhood" of a node refers to all directly connected nodes, meaning the immediate neighbors of that node. The "second neighborhood" extends this concept by considering the neighbors of those immediate neighbors.

Sumner's conjecture is a conjecture in graph theory proposed by the mathematician D.P. Sumner in 1981. It deals with the concept of graph embeddings and the existence of certain subgraphs within larger graphs.

Szymanski's conjecture refers to a problem in the field of number theory, particularly concerning prime numbers. Specifically, it conjectures the existence of infinitely many primes of a certain form related to the sequence of prime numbers. The conjecture states that for any integer \( n \geq 1 \), there are infinitely many primes of the form \( p_k = k^2 + n \) for some positive integer \( k \).