Erdős Problem 944

Reference: erdosproblems.com/944

def Erdos944.SimpleGraph.IsErdos944 {V : Type u} (G : SimpleGraph V) (k r : ℕ) : Prop

The predicate that graph $G$ with chromatic number $k$ is such that every vertex is critical, yet every critical set of edges has size $>r$

Equations

• Erdos944.SimpleGraph.IsErdos944 G k r = (G.IsCritical k ∧ ∀ (edges : Set (Sym2 V)), G.IsCriticalEdges edges → r < edges.ncard)

theorem Erdos944.erdos_944 {V : Type u} : (∀ k ≥ 4, ∀ r ≥ 1, ∃ (G : SimpleGraph V), SimpleGraph.IsErdos944 G k r) ↔ sorry

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 $>r$?

theorem Erdos944.erdos_944.variants.dirac_conjecture {V : Type u} : (∀ k ≥ 4, ∃ (G : SimpleGraph V), SimpleGraph.IsErdos944 G k 1) ↔ sorry

Let $k \ge 4$. Must there exist a graph $G$ with chromatic number $k$ such that every vertex is critical, yet every critical set of edges has size $>1$?

This was conjectured by Dirac in 1970.

theorem Erdos944.erdos_944.variants.dirac_conjecture.k_eq_5 {V : Type u} : ∃ (G : SimpleGraph V), SimpleGraph.IsErdos944 G 5 1

Dirac's conjecture was proved, for $k=5$: There exists a graph $G$ with chromatic number $5$, such that every vertex is critical, yet every critical set of edges has size $>1$, or in other words: has no critical edge.

[Br92] Brown, Jason I., A vertex critical graph without critical edges. Discrete Math. (1992), 99--101

theorem Erdos944.erdos_944.variants.dirac_conjecture.k_sub_one_not_prime {V : Type u} (k : ℕ) (hk : 4 ≤ k) (h : ¬Nat.Prime (k - 1)) : ∃ (G : SimpleGraph V), SimpleGraph.IsErdos944 G k 1

Lattanzio [La02] proved there exist $k$-critical graphs without critical edges for all $k$ such that $k - 1$ is not prime.

[La02] Lattanzio, John J., A note on a conjecture of {D}irac. Discrete Math. (2002), 323--330

theorem Erdos944.erdos_944.variants.dirac_conure.k_ge_five {V : Type u} (k : ℕ) (hk : 5 ≤ k) : ∃ (G : SimpleGraph V), SimpleGraph.IsErdos944 G k 1

Jensen [Je02] gave an construction for $k$-critical graphs without any critical edges for all $k≥5$.

[Je02] Jensen, Tommy R., Dense critical and vertex-critical graphs. Discrete Math. (2002), 63--84.

theorem Erdos944.erdos_944.dirac_conjecture.k_eq_four {V : Type u} : (∃ (G : SimpleGraph V), SimpleGraph.IsErdos944 G 4 1) ↔ sorry

The case $k=4$ and $r=1$ remains open: Are there $4$-critical graphs without any critical edges?

theorem Erdos944.erdos_944.variants.large_k_for_any_r {V : Type u} (r : ℕ) (hr : 1 ≤ r) : ∀ᶠ (k : ℕ) in Filter.atTop, ∃ (G : SimpleGraph V), SimpleGraph.IsErdos944 G k r

Martinsson and Steiner [MaSt25] proved for every $r \ge 1$ if $k$ is sufficiently large, depending on $r$, there exist a graph $G$ with chromatic number $k$ such that every vertex is critical, yet every critical set of edges has size $>r$.

[MaSt25] Martinsson, Anders and Steiner, Raphael, Vertex-critical graphs far from edge-criticality. Combin. Probab. Comput. (2025), 151--157