Peaceful Colourings
arxiv(2024)
摘要
We introduce peaceful colourings, a variant of h-conflict free colourings.
We call a colouring with no monochromatic edges p-peaceful if for each vertex
v, there are at most p neighbours of v coloured with a colour appearing
on another neighbour of v. An h-conflict-free colouring of a graph is a
(vertex)-colouring with no monochromatic edges so that for every vertex v,
the number of neighbours of v which are coloured with a colour appearing on
no other neighbour of v is at least the minimum of h and the degree of v.
If G is Δ-regular then it has an h-conflict free colouring precisely
if it has a (Δ-h)-peaceful colouring. We focus on the minimum p_Δ
of those p for which every graph of maximum degree Δ has a
p-peaceful colouring with Δ+1 colours. We show that p_Δ >
(1-1/e-o(1))Δ and that for graphs of bounded codegree, p_Δ≤ (1-1/e+o(1))Δ. We ask if the latter result can be improved
by dropping the bound on the codegree. As a partial result, we show that
p_Δ≤8000/8001Δ for sufficiently large Δ.
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