FormalConjectures.ErdosProblems.«583»

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module

public import FormalConjecturesUtil

Erdős Problem 583

References:

erdosproblems.com/583

[AnBa23] Anto, Nevil and Basavaraju, Manu, Gallai's path decomposition for 2-degenerate graphs. Discrete Math. Theor. Comput. Sci. (2023), Paper No. 16, 11.

[BBB21] A. Blanché, M. Bonamy, and N. Bonichon, Gallai's path decomposition in planar graphs. arXiv:2110.08870 (2021).

[BoPe19] Bonamy, Marthe and Perrett, Thomas J., Gallai's path decomposition conjecture for graphs of small maximum degree. Discrete Math. (2019), 1293--1299.

[CFZ26] Chu, Yanan and Fan, Genghua and Zhou, Chuixiang, Gallai's conjecture and the path number of odd semi-cliques. Discrete Math. (2026), Paper No. 114725, 6.

[Ch78] Chung, F. R. K., On partitions of graphs into trees. Discrete Math. (1978), 23-30.

[DeKo00] Dean, Nathaniel and Kouider, Mekkia, Gallai's conjecture for disconnected graphs. Discrete Math. (2000), 43--54.

[Er71] Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109.

[Fa02] Fan, Genghua, Subgraph coverings and edge switchings. J. Combin. Theory Ser. B (2002), 54-83.

[Lo68] Lovász, L., On covering of graphs. Theory of Graphs (Proc. Colloq., Tihany, 1966) (1968), 231-236.

[Py96] Pyber, L., Covering the edges of a connected graph by paths. J. Combin. Theory Ser. B (1996), 152-159.

@[expose] public sectionopen SimpleGraphnamespace Erdos583

A subgraph H of G is a path subgraph if it is the subgraph traced out by a path in G, i.e. a walk with no repeated vertices.

def IsPathSubgraph {V : Type*} {G : SimpleGraph V} (H : G.Subgraph) : Prop :=
  ∃ (u v : V) (p : G.Walk u v), p.IsPath ∧ H = p.toSubgraph

Every connected graph on $n$ vertices can be partitioned into at most $\lceil n/2\rceil$ edge-disjoint paths.

A problem of Erdős and Gallai.

@[category research open, AMS 5]
theorem erdos_583 {V : Type*} [Fintype V] (G : SimpleGraph V) (hG : G.Connected) :
    ∃ D : Finset G.Subgraph,
      (∀ H ∈ D, IsPathSubgraph H) ∧
      IsDecomposition G D ∧
      D.card ≤ ⌈(Fintype.card V : ℚ) / 2⌉₊ := byV:Type u_1inst✝:Fintype VG:SimpleGraph VhG:G.Connected⊢ ∃ D, (∀ H ∈ D, IsPathSubgraph H) ∧ G.IsDecomposition D ∧ D.card ≤ ⌈↑(Fintype.card V) / 2⌉₊
  sorryAll goals completed! 🐙

end Erdos583