Conjecture Hold Research Articles

Thin edges in cubic braces

AbstractFor a vertex set in a graph, the edge cut is the set of edges with exactly one end vertex in . An edge cut is tight if every perfect matching of the graph contains exactly one edge in . A matching covered bipartite graph is a brace if, for every tight cut , or , where . Braces play an important role in Lovász's tight cut decomposition of matching covered graphs. The bicontraction of a vertex of degree two in a graph, with precisely two neighbours and , consists of shrinking the set to a single vertex. The retract of a matching covered graph is the graph obtained from by repeatedly the bicontractions of vertices of degree two. An edge of a brace with at least six vertices is thin if the retract of is a brace. McCuaig showed that every brace of order at least six has a thin edge. In a brace of order six or more, Carvalho, Lucchesi and Murty proved that has two thin edges, and conjectured that contains two nonadjacent thin edges. Further, they made a stronger conjecture: There exists a positive constant such that every brace has thin edges. By showing that, in every cubic brace of order at least six, there exists a matching of size at least such that every edge in is thin, we prove that the above two conjectures hold for cubic braces.

Borodin–Kostochka Conjecture Holds for Odd-Hole-Free Graphs

Borodin–Kostochka Conjecture Holds for Odd-Hole-Free Graphs

Subrings of polynomial rings and the conjectures of Eisenbud and Evans

Let R be a commutative Noetherian ring of dimension d. In 1973, Eisenbud and Evans proposed three conjectures on the polynomial ring R[T]. These conjectures were settled in the affirmative by Sathaye, Mohan Kumar and Plumstead. One of the primary objectives of this article is to investigate the validity of these conjectures over Noetherian subrings of R[T] of dimension d+1, containing R. We formulate a class of such rings, which includes polynomial rings, Rees algebras, Rees-like algebras and Noetherian symbolic Rees algebras, and exhibit that all three conjectures hold for rings belonging to this class.

On Seymour's and Sullivan's second neighbourhood conjectures

AbstractFor a vertex of a digraph, (, respectively) is the number of vertices at distance 1 from (to, respectively) and is the number of vertices at distance 2 from . In 1995, Seymour conjectured that for any oriented graph there exists a vertex such that . In 2006, Sullivan conjectured that there exists a vertex in such that . We give a sufficient condition in terms of the number of transitive triangles for an oriented graph to satisfy Sullivan's conjecture. In particular, this implies that Sullivan's conjecture holds for all orientations of planar graphs and triangle‐free graphs. An oriented graph is an oriented split graph if the vertices of can be partitioned into vertex sets and such that is an independent set and induces a tournament. We also show that the two conjectures hold for some families of oriented split graphs, in particular, when induces a regular or an almost regular tournament.

Fuglede’s Conjecture Holds for Cyclic Groups of Order pqrs

The tile-spectral direction of the discrete Fuglede-conjecture is well-known for cyclic groups of square-free order, initiated by Łaba and Meyerowitz, but the spectral-tile direction is far from being well-understood. The product of at most three primes as the order of the cyclic group was studied intensely in the last couple of years. In this paper we study the case when the order of the cyclic group is the product of four different primes and prove that Fuglede’s conjecture holds in this case.

Bourgain’s slicing problem and KLS isoperimetry up to polylog

We prove that Bourgain’s hyperplane conjecture and the Kannan-Lovász-Simonovits (KLS) isoperimetric conjecture hold true up to a factor that is polylogarithmic in the dimension.

Equivalence classes and conditional hardness in massively parallel computations

The Massively Parallel Computation (MPC) model serves as a common abstraction of many modern large-scale data processing frameworks, and has been receiving increasingly more attention over the past few years, especially in the context of classical graph problems. So far, the only way to argue lower bounds for this model is to condition on conjectures about the hardness of some specific problems, such as graph connectivity on promise graphs that are either one cycle or two cycles, usually called the one cycle versus two cycles problem. This is unlike the traditional arguments based on conjectures about complexity classes (e.g., textsf {P}ne textsf {NP}), which are often more robust in the sense that refuting them would lead to groundbreaking algorithms for a whole bunch of problems. In this paper we present connections between problems and classes of problems that allow the latter type of arguments. These connections concern the class of problems solvable in a sublogarithmic amount of rounds in the MPC model, denoted by textsf {MPC}(o(log N)), and the standard space complexity classes textsf {L} and textsf {NL}, and suggest conjectures that are robust in the sense that refuting them would lead to many surprisingly fast new algorithms in the MPC model. We also obtain new conditional lower bounds, and prove new reductions and equivalences between problems in the MPC model. Specifically, our main results are as follows.Lower bounds conditioned on the one cycle versus two cycles conjecture can be instead argued under the textsf {L}nsubseteq textsf {MPC}(o(log N)) conjecture: these two assumptions are equivalent, and refuting either of them would lead to o(log N)-round MPC algorithms for a large number of challenging problems, including list ranking, minimum cut, and planarity testing. In fact, we show that these problems and many others require asymptotically the same number of rounds as the seemingly much easier problem of distinguishing between a graph being one cycle or two cycles.Many lower bounds previously argued under the one cycle versus two cycles conjecture can be argued under an even more robust (thus harder to refute) conjecture, namely textsf {NL}nsubseteq textsf {MPC}(o(log N)). Refuting this conjecture would lead to o(log N)-round MPC algorithms for an even larger set of problems, including all-pairs shortest paths, betweenness centrality, and all aforementioned ones. Lower bounds under this conjecture hold for problems such as perfect matching and network flow.

Fault-tolerant quantum speedup from constant depth quantum circuits

A defining feature in the field of quantum computing is the potential of a quantum device to outperform its classical counterpart for a specific computational task. By now, several proposals exist showing that certain sampling problems can be done efficiently quantumly, but are not possible efficiently classically, assuming strongly held conjectures in complexity theory. A feature dubbed quantum speedup. However, the effect of noise on these proposals is not well understood in general, and in certain cases it is known that simple noise can destroy the quantum speedup. Here we develop a fault-tolerant version of one family of these sampling problems, which we show can be implemented using quantum circuits of constant depth. We present two constructions, each taking $poly(n)$ physical qubits, some of which are prepared in noisy magic states. The first of our constructions is a constant depth quantum circuit composed of single and two-qubit nearest neighbour Clifford gates in four dimensions. This circuit has one layer of interaction with a classical computer before final measurements. Our second construction is a constant depth quantum circuit with single and two-qubit nearest neighbour Clifford gates in three dimensions, but with two layers of interaction with a classical computer before the final measurements. For each of these constructions, we show that there is no classical algorithm which can sample according to its output distribution in $poly(n)$ time, assuming two standard complexity theoretic conjectures hold. The noise model we assume is the so-called local stochastic quantum noise. Along the way, we introduce various new concepts such as constant depth magic state distillation (MSD), and constant depth output routing, which arise naturally in measurement based quantum computation (MBQC), but have no constant-depth analogue in the circuit model.

Graham's Pebbling Conjecture Holds for the Product of a Graph and a Sufficiently Large Complete Graph

For connected graphs $G$ and $H$, Graham conjectured that $\pi(G\square H)\leq\pi(G)\pi(H)$ where $\pi(G), \pi(H)$, and $\pi(G\square H)$ are the pebbling numbers of $G$, $H$, and the Cartesian product $G\square H$, respectively. In this paper, we show that the inequality holds when $H$ is a complete graph of sufficiently large order in terms of graph parameters of $G$.

The strong Atiyah and Lück approximation conjectures for one-relator groups

It is shown that the strong Atiyah conjecture and the L\"uck approximation conjecture in the space of marked groups hold for locally indicable groups. In particular, this implies that one-relator groups satisfy both conjectures. We also show that the center conjecture, the independence conjecture and the strong eigenvalue conjecture hold for these groups. As a byproduct we prove that the group algebra of a locally indicable group over a field of characteristic zero has a Hughes-free epic division algebra and, in particular, it is embedded in a division algebra.

Quasilinear quadratic forms and function fields of quadrics

Let p and q be anisotropic quadratic forms of dimension $$\ge 2$$ over a field F. In a recent article, we formulated a conjecture describing a general constraint which the dimensions of p and q impose on the isotropy index of q after scalar extension to the function field of p. This can be viewed as a generalization of Hoffmann’s Separation Theorem which simultaneously incorporates and refines some well-known classical results on the Witt kernels of function fields of quadrics. Using algebro-geometric methods, it was shown that large parts of this conjecture hold in the case where the characteristic of F is not 2. In the present article, we prove similar (in fact, somewhat sharper) results in the case where F has characteristic 2 and q is a so-called quasilinear form. In contrast to the situation where $$\mathrm {char}(F) \ne 2$$, the methods used to treat this case are purely algebraic.

The base change in the Atiyah and the Lück approximation conjectures

Let F be a free finitely generated group and $${A \in {\rm Mat}_{n \times m}(\mathbb{C}[F])}$$ . For each quotient G = F/N of F we can define a von Neumann rank function rkG(A) associated with the l2-operator l2(G)n → l2(G)m induced by right multiplication by A. For example, in the case where G is finite, $${{\rm rk}_G(A)=\frac{{\rm rk}_{\mathbb{C}}({{\bar{A}}})}{|G|}}$$ is the normalized rank of the matrix $${{{\bar{A}}} \in {\rm Mat}_{n \times m}(\mathbb{C}[G])}$$ obtained by reducing the coefficients of A modulo N. One of the variations of the Luck approximation conjecture claims that the function $${N\mapsto {\rm rk}_{F/N}(A)}$$ is continuous in the space of marked groups. The strong Atiyah conjecture predicts that if the least common multiple lcm(G) of the orders of finite subgroups of G is finite, then $${{\rm rk}_G(A) \in \frac{1}{lcm (G)}\mathbb{Z}}$$ . In our first result we prove the sofic Luck approximation conjecture. In particular, we show that the function $${N \mapsto {\rm rk}_{F/N}(A)}$$ is continuous in the space of sofic marked groups. Among other consequences we obtain that a strong version of the algebraic eigenvalue conjecture, the center conjecture and the independence conjecture hold for sofic groups. In our second result we apply the sofic Luck approximation and we show that the strong Atiyah conjecture holds for groups from a class $${{\mathcal{D}}}$$ , virtually compact special groups, Artin’s braid groups and torsion-free p-adic analytic pro-p groups.

Finite groups with odd Sylow automizers

Let p be an odd prime number. In this paper, we characterize the nonabelian composition factors of a finite group with odd p-Sylow automizers, and then prove that the McKay conjecture, the Alperin weight conjecture, and the Alperin–McKay conjecture hold for such a group.

The convex set containing two-qutrit maximally entangled states

We construct the convex set $$\mathcal{M}$$ of two-qutrit states, and the subset $$\mathcal{P}\subset \mathcal{M}$$ . We characterize the extremal points of rank one and rank two of $$\mathcal{M}$$ . We further show that the extremal points of $$\mathcal{P}$$ have rank not equaling to two, and at most six. We apply our results to a long-standing conjecture on the locally distinguishable subspace under local operations and classical communications. We prove that rank-one or rank-two matrices in $$\mathcal{M}$$ hold for the conjecture. We further simplify the conjecture, by showing that the conjecture holds if and only if it holds for states in $$\mathcal{P}$$ .

Enomoto and Ota’s Conjecture Holds for Large Graphs

In 2000, Enomoto and Ota conjectured that if a graph G satisfies $$\sigma _{2}(G) \ge |G| + k - 1$$ , then for any set of k vertices $$v_{1}, \ldots , v_{k}$$ and for any positive integers $$n_{1}, \ldots , n_{k}$$ with $$\sum n_{i} = |G|$$ , there exists a partition of V(G) into k paths $$P_{1}, \ldots , P_{k}$$ such that $$v_{i}$$ is an end of $$P_{i}$$ and $$|P_{i}| = n_{i}$$ for all i. We prove this conjecture when |G| is large. Our proof uses the Regularity Lemma along with several extremal lemmas, concluding with an absorbing argument to retrieve misbehaving vertices.

Some bounds on the number of colors in interval and cyclic interval edge colorings of graphs

An intervalt-coloring of a multigraph G is a proper edge coloring with colors 1,…,t such that the colors of the edges incident with every vertex of G are colored by consecutive colors. A cyclic intervalt-coloring of a multigraph G is a proper edge coloring with colors 1,…,t such that the colors of the edges incident with every vertex of G are colored by consecutive colors, under the condition that color 1 is considered as consecutive to color t. Denote by w(G) (wc(G)) and W(G) (Wc(G)) the minimum and maximum number of colors in a (cyclic) interval coloring of a multigraph G, respectively. We present some new sharp bounds on w(G) and W(G) for multigraphs G satisfying various conditions. In particular, we show that if G is a 2-connected multigraph with an interval coloring, then W(G)≤1+|V(G)|2(Δ(G)−1). We also give several results towards the general conjecture that Wc(G)≤|V(G)| for any triangle-free graph G with a cyclic interval coloring; we establish that approximate versions of this conjecture hold for several families of graphs, and we prove that the conjecture is true for graphs with maximum degree at most 4.

A Counterexample to the “Hot Spots” Conjecture on Nested Fractals

Although the “hot spots” conjecture was proved to be false on some classical domains, the problem still generates a lot of interests on identifying the domains that the conjecture hold. The question can also be asked on fractal sets that admit Laplacians. It is known that the conjecture holds on the Sierpinski gasket and its variants. In this note, we show surprisingly that the “hot spots” conjecture fails on the hexagasket, a typical nested fractal set. The technique we use is the spectral decimation method of eigenvalues of Laplacian on fractals.

Refinements of Dade's Projective Conjecture for p-solvable groups

Dade's Projective Conjecture is known to be true for finite p-solvable groups thanks to work of G.R. Robinson, but remains open in general. Work of Isaacs and Navarro suggested to Uno and Boltje refinements of this conjecture. These refinements were studied for finite p-solvable groups by Glesser. In the present paper, inspired by earlier work of Turull, we propose further refinements of the conjecture that take into account the Schur indices and the elements of the Brauer group. We prove that all these refinements of Dade's Projective Conjecture hold for all finite p-solvable groups. In particular, we obtain that the version of Dade's Projective Conjecture which involves character degree residues modulo p, fields of definition and Schur indices, as well as the full strength of Boltje's Conjecture both hold for all finite p-solvable groups. The proof develops a Clifford theory for normalizers of chains of p-subgroups which allows one to reduce the calculation of the relevant sums to simpler groups.

Neighbor distinguishing total choice number of sparse graphs via the Combinatorial Nullstellensatz

Let G = (V,E) be a graph and ϕ: V ∪E → {1, 2, · · ·, k} be a total-k-coloring of G. Let f(v)(S(v)) denote the sum(set) of the color of vertex v and the colors of the edges incident with v. The total coloring ϕ is called neighbor sum distinguishing if (f(u) ≠ f(v)) for each edge uv ∈ E(G). We say that ϕ is neighbor set distinguishing or adjacent vertex distinguishing if S(u) ≠ S(v) for each edge uv ∈ E(G). For both problems, we have conjectures that such colorings exist for any graph G if k ≥ Δ(G) + 3. The maximum average degree of G is the maximum of the average degree of its non-empty subgraphs, which is denoted by mad (G). In this paper, by using the Combinatorial Nullstellensatz and the discharging method, we prove that these two conjectures hold for sparse graphs in their list versions. More precisely, we prove that every graph G with maximum degree Δ(G) and maximum average degree mad(G) has chΣ″(G) ≤ Δ(G) + 3 (where chΣ″(G) is the neighbor sum distinguishing total choice number of G) if there exists a pair \((k,m) \in \{ (6,4),(5,\tfrac{{18}} {5}),(4,\tfrac{{16}} {5})\}\) such that Δ(G) ≥ k and mad (G) <m.

The dynamics of semigroups of transcendental entire functions II

We introduce the concept of escaping set for semigroups of transcendental entire functions using Fatou-Julia theory. Several results of the escaping set associated with the iteration of one transcendental entire function have been extended to transcendental semigroups. We also investigate the properties of escaping sets for conjugate semigroups and abelian transcendental semigroups. Several classes of transcendental semigroups for which Eremenko's conjectures hold have been provided.