Erdős Problem #944

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

Deleting a vertex also deletes its incident edges; deleting an edge set leaves the vertices in place. The parameters $k,r$ are fixed before constructing $G$. The two types of criticality in the question must not be interchanged.

A graph has a vertex set $V$ and edges that are unordered pairs of distinct vertices. A subgraph need not be induced unless that word is stated. A proper vertex colouring gives adjacent vertices different colours; $\chi(G)$ is the least number of colours. [S1] [S2]

Statement recorded in the supplied source.

A critical vertex, edge, or set of edges, is one whose deletion lowers the chromatic number. Let $k\geq 4$ and $r\geq 1$. Must there exist a graph $G$ with chromatic number $k$ such that every vertex is critical, yet every critical set of edges has size $>r$? [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Resolve k=4, already open for r=1. [S1]

Short English statement

Can fixed-chromatic graphs have every vertex essential to their chromatic number while no bounded-size edge deletion reduces it?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2398 (EP-944), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 T. F. Bloom, Erdős Problems, Problem 944, https://www.erdosproblems.com/944. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Brown, Jason I., A vertex critical graph without critical edges. Discrete Math. (1992), 99--101. Bibliographic information from the supplied record.

• S4 Jensen, Tommy R., Dense critical and vertex-critical graphs. Discrete Math. (2002), 63--84. Bibliographic information from the supplied record.

• S5 Lattanzio, John J., A note on a conjecture of {D}irac. Discrete Math. (2002), 323--330. Bibliographic information from the supplied record.

• S6 Martinsson, Anders and Steiner, Raphael, Vertex-critical graphs far from edge-criticality. Combin. Probab. Comput. (2025), 151--157. Bibliographic information from the supplied record.