1 research outputs found
A structure of 1-planar graph and its applications to coloring problems
A graph is 1-planar if it can be drawn on a plane so that each edge is
crossed by at most one other edge. In this paper, we first give a useful
structural theorem for 1-planar graphs, and then apply it to the list edge and
list total coloring, the -total labelling, and the equitable edge
coloring of 1-planar graphs. More precisely, we verify the well-known List Edge
Coloring Conjecture and List Total Coloring Conjecture for 1-planar graph with
maximum degree at least 18, prove that the -total labelling number of
every 1-planar graph is at most provided that
and , and show that every 1-planar graph has an
equitable edge coloring with colors for any integer . These three
results respectively generalize the main theorems of three different previously
published papers.Comment: 13 page