Erdős Problem #149

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.

An induced matching is a set of vertex-disjoint edges with no further edge between their endpoints. A strong edge colouring partitions $E(G)$ into induced matchings. This is stricter than a proper edge colouring, which only separates edges sharing a vertex.

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.

Let $G$ be a graph with maximum degree $\Delta$. Is $G$ the union of at most $\tfrac{5}{4}\Delta^2$ sets of strongly independent edges (sets such that the induced subgraph is the union of vertex-disjoint edges)? [S1]

Short English statement

Can every graph's edges be split into induced matchings using at most five quarters of the square of its maximum degree?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 1958 (EP-149), 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 149, https://www.erdosproblems.com/149. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Eoin Hurley, Remi de Joannis de Verclos, and Ross J. Kang, An improved procedure for colouring graphs of bounded local density, Advances in Combinatorics (2022), Paper 7,

  \nolinkurl{doi:10.19086/aic.2022.7.}

  https://arxiv.org/abs/2007.07874. Bibliographic information from the supplied record.

• S4 Wouter Cames van Batenburg, Ross J. Kang, and Francois Pirot, Strong cliques and forbidden cycles, Indagationes Mathematicae 31 (2020), 64-82,

  \nolinkurl{doi:10.1016/j.indag.2019.09.003.}

  https://arxiv.org/abs/1903.06087. Bibliographic information from the supplied record.

• S5 Bonamy, Marthe and Perrett, Thomas and Postle, Luke, Colouring graphs with sparse neighbourhoods: bounds and applications. J. Combin. Theory Ser. B (2022), 278-317. Bibliographic information from the supplied record.

• S6 Bruhn, Henning and Joos, Felix, A stronger bound for the strong chromatic index. Combin. Probab. Comput. (2018), 21-43. Bibliographic information from the supplied record.