Displaying similar documents to “Decomposing a planar graph into a forest and a subgraph of restricted maximum degree.”

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

Xin Zhang, Guizhen Liu (2013)

Open Mathematics

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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...

Note: The Smallest Nonevasive Graph Property

Michał Adamaszek (2014)

Discussiones Mathematicae Graph Theory

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A property of n-vertex graphs is called evasive if every algorithm testing this property by asking questions of the form “is there an edge between vertices u and v” requires, in the worst case, to ask about all pairs of vertices. Most “natural” graph properties are either evasive or conjectured to be such, and of the few examples of nontrivial nonevasive properties scattered in the literature the smallest one has n = 6. We exhibit a nontrivial, nonevasive property of 5-vertex graphs...

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

Xin Zhang, Yong Yu, Guizhen Liu (2011)

Open Mathematics

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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.

Edge-disjoint odd cycles in graphs with small chromatic number

Claude Berge, Bruce Reed (1999)

Annales de l'institut Fourier

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For a simple graph, we consider the minimum number of edges which block all the odd cycles and the maximum number of odd cycles which are pairwise edge-disjoint. When these two coefficients are equal, interesting consequences appear. Similar problems (but interchanging “ odd cycles” and “ odd cycles”) have been considered in a paper by Berge and Fouquet.