- Award ID(s):
- 1950189
- NSF-PAR ID:
- 10209860
- Date Published:
- Journal Name:
- Mathematics
- Volume:
- 9
- Issue:
- 1
- ISSN:
- 2227-7390
- Page Range / eLocation ID:
- 1-16
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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null (Ed.)A graph G is called {\em self-ordered} (a.k.a asymmetric) if the identity permutation is its only automorphism. Equivalently, there is a unique isomorphism from G to any graph that is isomorphic to G. We say that G=(VE) is {\em robustly self-ordered}if the size of the symmetric difference between E and the edge-set of the graph obtained by permuting V using any permutation :VV is proportional to the number of non-fixed-points of . In this work, we initiate the study of the structure, construction and utility of robustly self-ordered graphs. We show that robustly self-ordered bounded-degree graphs exist (in abundance), and that they can be constructed efficiently, in a strong sense. Specifically, given the index of a vertex in such a graph, it is possible to find all its neighbors in polynomial-time (i.e., in time that is poly-logarithmic in the size of the graph). We provide two very different constructions, in tools and structure. The first, a direct construction, is based on proving a sufficient condition for robust self-ordering, which requires that an auxiliary graph, on {\em pairs} of vertices of the original graph, is expanding. In this case the original graph is (not only robustly self-ordered but) also expanding. The second construction proceeds in three steps: It boosts the mere existence of robustly self-ordered graphs, which provides explicit graphs of sublogarithmic size, to an efficient construction of polynomial-size graphs, and then, repeating it again, to exponential-size(robustly self-ordered) graphs that are locally constructible. This construction can yield robustly self-ordered graphs that are either expanders or highly disconnected, having logarithmic size connected components. We also consider graphs of unbounded degree, seeking correspondingly unbounded robustness parameters. We again demonstrate that such graphs (of linear degree)exist (in abundance), and that they can be constructed efficiently, in a strong sense. This turns out to require very different tools. Specifically, we show that the construction of such graphs reduces to the construction of non-malleable two-source extractors with very weak parameters but with some additional natural features. We actually show two reductions, one simpler than the other but yielding a less efficient construction when combined with the known constructions of extractors. We demonstrate that robustly self-ordered bounded-degree graphs are useful towards obtaining lower bounds on the query complexity of testing graph properties both in the bounded-degree and the dense graph models. Indeed, their robustness offers efficient, local and distance preserving reductions from testing problems on ordered structures (like sequences) to the unordered (effectively unlabeled) graphs. One of the results that we obtain, via such a reduction, is a subexponential separation between the query complexities of testing and tolerant testing of graph properties in the bounded-degree graph model. Changes to previous version: We retract the claims made in our initial posting regarding the construction of non-malleable two-source extractors (which are quasi-orthogonal) as well as the claims about the construction of relocation-detecting codes (see Theorems 1.5 and 1.6 in the original version). The source of trouble is a fundamental flaw in the proof of Lemma 9.7 (in the original version), which may as well be wrong. Hence, the original Section 9 was omitted, except that the original Section 9.3 was retained as a new Section 8.3. The original Section 8 appears as Section 8.0 and 8.1, and Section 8.2 is new.more » « less
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Abstract Let
be integers with , and set . Erdős proved that when , each n ‐vertex nonhamiltonian graphG with minimum degreehas at most edges. He also provides a sharpness example for all such pairs . Previously, we showed a stability version of this result: for n large enough, every nonhamiltonian graphG onn vertices withand more than edges is a subgraph of . In this article, we show that not only does the graph maximize the number of edges among nonhamiltonian graphs with n vertices and minimum degree at leastd , but in fact it maximizes the number of copies of any fixed graphF whenn is sufficiently large in comparison withd and. We also show a stronger stability theorem, that is, we classify all nonhamiltonian n ‐vertex graphs withand more than edges. We show this by proving a more general theorem: we describe all such graphs with more than copies of for any k . -
Given any graph G G , the spread of G G is the maximum difference between any two eigenvalues of the adjacency matrix of G G . In this paper, we resolve a pair of 20-year-old conjectures of Gregory, Hershkowitz, and Kirkland regarding the spread of graphs. The first states that for all positive integers n n , the n n -vertex graph G G that maximizes spread is the join of a clique and an independent set, with ⌊ 2 n / 3 ⌋ \lfloor 2n/3 \rfloor and ⌈ n / 3 ⌉ \lceil n/3 \rceil vertices, respectively. Using techniques from the theory of graph limits and numerical analysis, we prove this claim for all n n sufficiently large. As an intermediate step, we prove an analogous result for a family of operators in the Hilbert space over L 2 [ 0 , 1 ] \mathscr {L}^2[0,1] . The second conjecture claims that for any fixed m ≤ n 2 / 4 m \leq n^2/4 , if G G maximizes spread over all n n -vertex graphs with m m edges, then G G is bipartite. We prove an asymptotic version of this conjecture. Furthermore, we construct an infinite family of counterexamples, which shows that our asymptotic solution is tight up to lower-order error terms.more » « less
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Abstract A graph is ‐
free if it has no induced subgraph isomorphic to , and |G | denotes the number of vertices of . A conjecture of Conlon, Sudakov and the second author asserts that:For every graph , there exists such that in every ‐free graph with |
G | there are two disjoint sets of vertices, of sizes at least and , complete or anticomplete to each other.The “sparse linear conjecture”: For every graph , there exists such that in every ‐free graph with , either some vertex has degree at least , or there are two disjoint sets of vertices, of sizes at least and , anticomplete to each other.
The sparse linear conjecture holds for all almost‐bipartite graphs .
For every almost‐bipartite graph , there exist such that for every graph with and maximum degree less than , and for every with , either contains induced copies of , or there are two disjoint sets with and , and with at most edges between them.
For every graph , there exists such that in every ‐free graph with vertices, either some vertex has degree at least , or there are two disjoint sets of vertices with , anticomplete to each other.
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Abstract Bootstrap percolation is a deterministic cellular automaton in which vertices of a graph begin in one of two states, “dormant” or “active.” Given a fixed positive integer , a dormant vertex becomes active if at any stage it has at least active neighbors, and it remains active for the duration of the process. Given an initial set of active vertices , we say that
G ‐percolates (from ) if every vertex in becomes active after some number of steps. Let denote the minimum size of a set such thatG ‐percolates from . Bootstrap percolation has been studied in a number of settings and has applications to both statistical physics and discrete epidemiology. Here, we are concerned with degree‐based density conditions that ensure . In particular, we give an Ore‐type degree sum result that states that if a graph satisfies , then either or is in one of a small number of classes of exceptional graphs. (Here, is the minimum sum of degrees of two nonadjacent vertices in .) We also give a Chvátal‐type degree condition: If is a graph with degree sequence such that or for all , then or falls into one of several specific exceptional classes of graphs. Both of these results are inspired by, and extend, an Ore‐type result in a paper by Freund et al.