منابع مشابه
Sparse universal graphs for bounded-degree graphs
Let H be a family of graphs. A graph T is H-universal if it contains a copy of each H ∈ H as a subgraph. Let H(k, n) denote the family of graphs on n vertices with maximum degree at most k. For all positive integers k and n, we construct an H(k, n)-universal graph T with Ok(n 2 k log 4 k n) edges and exactly n vertices. The number of edges is almost as small as possible, as Ω(n2−2/k) is a lower...
متن کاملSparse Partition Universal Graphs for Graphs of Bounded Degree
In 1983, Chvátal, Trotter and the two senior authors proved that for any ∆ there exists B so that, for any n, any 2-coloring of the edges of the complete graph KN with N ≥ Bn vertices yields a monochromatic copy of any graph H that has n vertices and maximum degree ∆. We prove that the complete graph may be replaced by a sparser graph G that has N vertices and O(N2−1/∆ log N) edges, with N = bB...
متن کاملNear-Optimal Induced Universal Graphs for Bounded Degree Graphs
A graph U is an induced universal graph for a family F of graphs if every graph in F is a vertex-induced subgraph of U . For the family of all undirected graphs on n vertices Alstrup, Kaplan, Thorup, and Zwick [STOC 2015] give an induced universal graph with O ( 2n/2 ) vertices, matching a lower bound by Moon [Proc. of Glasgow Math. Association 1965]. Let k = dD/2e. Improving asymptotically on ...
متن کاملInduced-Universal Graphs for Graphs with Bounded Maximum Degree
For a family F of graphs, a graph U is induced-universal for F if every graph in F is an induced subgraph of U . We give a construction for an induceduniversal graph for the family of graphs on n vertices with degree at most r, which has Cnb(r+1)/2c vertices and Dn2b(r+1)/2c−1 edges, where C and D are constants depending only on r. This construction is nearly optimal when r is even in that such...
متن کاملOptimal induced universal graphs for bounded-degree graphs
We show that for any constant ∆ ≥ 2, there exists a graph Γ with O(n) vertices which contains every n-vertex graph with maximum degree ∆ as an induced subgraph. For odd ∆ this significantly improves the best-known earlier bound of Esperet et al. and is optimal up to a constant factor, as it is known that any such graph must have at least Ω(n) vertices. Our proof builds on the approach of Alon a...
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ژورنال
عنوان ژورنال: Random Structures and Algorithms
سال: 2007
ISSN: 1042-9832,1098-2418
DOI: 10.1002/rsa.20143