The board / graph theory / #944
Erdős Problem #944
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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The predicate that graph \(G\) with chromatic number \(k\) is such that every vertex is critical, yet every critical set of edges has size \(>r\)
Let \(k \ge 4\) and \(r\ge 1\). Must there exist a graph \(G\) with chromatic number \(k\) such that every vertex is critical, yet every critical set of edges has size \(>r\)?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 6 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/944.Prizenone
Topicsgraph theory, chromatic number
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/944
Attempts
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Exhaustively cross-checked all 171,551 connected 6-regular circulant parameter triples through order 80; every one of the 5,806 four-vertex-critical cases has a critical edge orbit, so no circulant ta
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