Group vertex-arboricity of group-labelled graphs

초록

We introduce the vertex-arboricity of group-labelled graphs. For an abelian group Γ, a Γ-labelled graph is a graph whose edges are labelled by elements of Γ. For an abelian group Γ and A⊆Γ, the (Γ,A)-vertex-arboricity of a Γ-labelled graph is the minimum integer k such that its vertex set can be partitioned into k parts where each part induces a subgraph having no cycle with γ -value in A . We prove that for every positive integer ω , there is a function fω:N×N→R such that if |Γ∖A|≤ω, then every Γ-labelled graph with (Γ,A)-vertex-arboricity at least fω(t,d) contains a subdivision of Kt where all branching paths have γ -value in A and length at least d . This extends the result that every graph of sufficiently large chromatic number contains a subdivision of Kt.

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