Sparse universal graphs for bounded‐degree graphs

Abstract

ABSTRACT: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 O k (n 2− 2 k log 4 k n) edges and exactly n vertices. The number of edges is almost as small as possible, as (n 2−2/k ) is a lower bound for the number of edges in any such graph. The construction of T is explicit, whereas the proof of universality is probabilistic and is based on a novel graph decomposition result and on the properties of random walks on expanders.

Powers of Hamilton cycles in pseudorandom graphs

“…It is worth noting that Alon and Capalbo [5] explicitly constructed almost optimally sparse universal graphs for spanning graphs with maximum degree ∆. These graphs have some pseudorandomness properties, but they also contain cliques of order log 2 n, which random graphs of the same density certainly do not.…”

“…Similarly, using goodness with respect to b 1 b 2 , we fix b 5 and b 8 . This readily implies by the definition of the candidate sets that (a 8 , a 7 , b 8 , b 7 )) forms a 2-path and that a 5 is adjacent to a 7 and a 8 (and to a 3 ), and similarly for b 5 . Next, by the definition of A 7 (a 5 ) we have that a 7 has a neighbour a 6 in A 6 (a 5 ), which we fix.…”

Discrete Kakeya-type problems and small bases

“…The problem has also been studied for other families of graphs, such as graphs of bounded degree, trees or planar graphs. The interested reader is referred to [2], [3] and the references therein.…”

Optimal induced universal graphs for bounded-degree graphs

“…(A graph is universal for a family H if it contains every H ∈ H as a subgraph (not necessarily induced)). The latter was studied in a series of papers [4,6] culminating in the work of the first author and Capalbo [5] where it is shown that this minimum is Θ(n 2−2/∆ ). Induced universal graphs for bounded-degree graphs were first studied by Butler [11].…”

“…In Section 3 we introduce the main building block in our construction, high-girth expander graphs, and state some of their properties. In Section 3.2 we state the decomposition result from [4], an analogue of Petersen's theorem for (2k + 1)-regular graphs. The construction of Γ is given in Section 4 and in Section 5 we prove that Γ is indeed induced universal.…”