منابع مشابه
Boxicity and maximum degree
An axis-parallel d–dimensional box is a Cartesian product R1 × R2 × · · · × Rd where Ri (for 1 ≤ i ≤ d) is a closed interval of the form [ai, bi] on the real line. For a graph G, its boxicity box(G) is the minimum dimension d, such that G is representable as the intersection graph of (axis–parallel) boxes in d–dimensional space. The concept of boxicity finds applications in various areas such a...
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The boxicity of a graph G = (V,E) is the smallest k for which there exist k interval graphs Gi = (V,Ei), 1 ≤ i ≤ k, such that E = E1 ∩ . . . ∩ Ek. Graphs with boxicity at most d are exactly the intersection graphs of (axis-parallel) boxes in Rd. In this note, we prove that graphs with maximum degree ∆ have boxicity at most ∆2 + 2, which improves the previous bound of 2∆2 obtained by Chandran et...
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A graph H is strongly immersed in G if G is obtained from H by a sequence of vertex splittings (i.e., lifting some pairs of incident edges and removing the vertex) and edge removals. Equivalently, vertices of H are mapped to distinct vertices of G (branch vertices) and edges of H are mapped to pairwise edge-disjoint paths in G, each of them joining the branch vertices corresponding to the ends ...
متن کاملVertex arboricity and maximum degree
The vertex arboricity of graph G is the minimum number of colors that can be used to color the vertices of G so that each color class induces an acyclic subgraph of G. We prove results such as this: if a connected graph G is neither a cycle nor a clique, then there is a coloring of V(G/ with at most [-A(G)/2 ~ colors, such that each color class induces a forest and one of those induced forests ...
متن کاملIntersection Dimension and Maximum Degree
We show that the intersection dimension of graphs with respect to several hereditary graph classes can be bounded as a function of the maximum degree. As an interesting special case, we show that the circular dimension of a graph with maximum degree ∆ is at most O(∆ log∆ log log∆). We also obtain bounds in terms of treewidth.
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ژورنال
عنوان ژورنال: Journal of Combinatorial Theory, Series B
سال: 2008
ISSN: 0095-8956
DOI: 10.1016/j.jctb.2007.08.002