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The structure of plane graphs with independent crossings and its applications to coloring problems

100%

Zhang X., Liu G.

Open Mathematics

2013

tom 11

nr 2

308-321

If a graph G has a drawing in the plane in such a way that every two crossings are independent, then we call G a plane graph with independent crossings or IC-planar graph for short. In this paper, the structure of IC-planar graphs with minimum degree at least two or three is studied. By applying their structural results, we prove that the edge chromatic number of G is Δ if Δ ≥ 8, the list edge (resp. list total) chromatic number of G is Δ (resp. Δ + 1) if Δ ≥ 14 and the linear arboricity of G is ℈Δ/2⌊ if Δ ≥ 17, where G is an IC-planar graph and Δ is the maximum degree of G.

On (p, 1)-total labelling of 1-planar graphs

100%

Zhang X., Yu Y., Liu G.

Open Mathematics

2011

tom 9

nr 6

1424-1434

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is proved that the (p, 1)-total labelling number of every 1-planar graph G is at most Δ(G) + 2p − 2 provided that Δ(G) ≥ 8p+4 or Δ(G) ≥ 6p+2 and g(G) ≥ 4. As a consequence, the well-known (p, 1)-total labelling conjecture has been confirmed for some 1-planar graphs.

Ograniczanie wyników

2 Open Mathematics

2 Liu G.

2 Zhang X.

1 Yu Y.

1 2013

1 2011

Wyniki wyszukiwania

The structure of plane graphs with independent crossings and its applications to coloring problems

On (p, 1)-total labelling of 1-planar graphs