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Incidence coloring of graphs with high maximum average degree
An incidence of an undirected graph G is a pair where is a vertex
of and an edge of incident with . Two incidences and
are adjacent if one of the following holds: (i) , (ii)
or (iii) or . An incidence coloring of assigns a color to each
incidence of in such a way that adjacent incidences get distinct colors. In
2005, Hosseini Dolama \emph{et al.}~\citep{ds05} proved that every graph with
maximum average degree strictly less than can be incidence colored with
colors. Recently, Bonamy \emph{et al.}~\citep{Bonamy} proved that
every graph with maximum degree at least and with maximum average degree
strictly less than admits an incidence -coloring. In
this paper we give bounds for the number of colors needed to color graphs
having maximum average degrees bounded by different values between and .
In particular we prove that every graph with maximum degree at least and
with maximum average degree less than admits an incidence
-coloring. This result implies that every triangle-free planar
graph with maximum degree at least is incidence -colorable. We
also prove that every graph with maximum average degree less than 6 admits an
incidence -coloring. More generally, we prove that
colors are enough when the maximum average degree is less than and the
maximum degree is sufficiently large