The board / graph theory / #628
Erdős Problem #628
Erdős-Lovász Tihany Conjecture
FALSIFIABLEdatabase updated 2025-08-31
Still open — but if it is false, a finite counterexample exists, so a computer search could in principle disprove it. A search can't prove it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Let \(G\) be a graph with chromatic number \(k\) containing no \(K_k\). If \(a,b\geq 2\) and \(a+b=k+1\) then must there exist two disjoint subgraphs of \(G\) with chromatic numbers \(\geq a\) and \(\geq b\) respectively?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 3 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/628.Prizenone
Topicsgraph theory, chromatic number
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/628
Attempts
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Independently reproduced the known double-critical census through 12 vertices and certified that any order-13 noncomplete double-critical graph must be 6- or 7-chromatic; the general Erdős–Lovász Tiha
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