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Let N k (m , n) denote the number of partitions of n with Garvan k -rank m. It is well-known that Andrews–Garvan–Dyson's crank and Dyson's rank are the k -rank for k = 1 and k = 2 , respectively. In this paper, we prove that the sequences (N k (m , n)) | m | ≤ n − k − 71 are log-concave for all sufficiently large integers n and each integer k. In particular, we partially solve the log-concavity conjecture for Andrews–Garvan–Dyson's crank and Dyson's rank, which was independently proposed by Bringmann–Jennings-Shaffer–Mahlburg and Ji–Zang recently. [ABSTRACT FROM AUTHOR]

In 1927, Artin conjectured that any integer a which is not −1 or a perfect square is a primitive root for a positive density of primes p. While this conjecture still remains open, there has been a lot of progress in last six decades. In 2000, Moree and Stevenhagen proposed what is known as the two variable Artin's conjecture and proved that for any multiplicatively independent rational numbers a and b , the set { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } has positive density under the Generalised Riemann Hypothesis for certain Dedekind zeta functions. While the infinitude of such primes is known, the only unconditional lower bound for the size of the above set is due to Ram Murty, Séguin and Stewart who in 2019 showed that for infinitely many pairs (a , b) # { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } ≫ x log 2 ⁡ x. In this paper we improve this lower bound. In particular we show that given any three multiplicatively independent integers S = { m 1 , m 2 , m 3 } such that m 1 , m 2 , m 3 , − 3 m 1 m 2 , − 3 m 2 m 3 , − 3 m 1 m 3 , m 1 m 2 m 3 are not squares, there exists a pair of elements a , b ∈ S such that # { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } ≫ x log ⁡ log ⁡ x log 2 ⁡ x. Further, under the assumption of a level of distribution greater than x 2 3 in a theorem of Bombieri, Friedlander and Iwaniec (as modified by Heath-Brown), we prove the following conditional result. Given any two multiplicatively independent integers S = { m 1 , m 2 } such that m 1 , m 2 , − 3 m 1 m 2 are not squares, there exists a pair of elements a , b ∈ { m 1 , m 2 , − 3 m 1 m 2 } such that # { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } ≫ x log ⁡ log ⁡ x log 2 ⁡ x. [ABSTRACT FROM AUTHOR]

Let k , t be coprime integers, and let 1 ≤ r ≤ t. We let D k × (r , t ; n) denote the total number of parts among all k -indivisible partitions (i.e., those partitions where no part is divisible by k) of n which are congruent to r modulo t. In previous work of the authors [3] , an asymptotic estimate for D k × (r , t ; n) was shown to exhibit unpredictable biases between congruence classes. In the present paper, we confirm our earlier conjecture in [3] that there are no "ties" (i.e., equalities) in this asymptotic for different congruence classes. To obtain this result, we reframe this question in terms of L -functions, and we then employ a nonvanishing result due to Baker, Birch, and Wirsing [1] to conclude that there is always a bias towards one congruence class or another modulo t among all parts in k -indivisible partitions of n as n becomes large. [ABSTRACT FROM AUTHOR]

This paper introduces the concept of infinite loops mod n and discusses their properties. In particular, it describes how the continued fraction expansions of infinite loops behave poorly under multiplication by the integer n. Infinite loops are geometric in origin, arising from viewing continued fractions as cutting sequences in the hyperbolic plane, however, they also have a nice description in terms of Diophantine approximation: An infinite loop mod n is any real number which has no semi-convergents divisible by n. The main result of this paper is a reformulation of the p -adic Littlewood Conjecture (pLC) in terms of infinite loops. More explicitly, this paper shows that a real number α is a counterexample to pLC if and only if there is some m ∈ N such that p ℓ α is an infinite loop mod p m , for all ℓ ∈ N ∪ { 0 }. [ABSTRACT FROM AUTHOR]

For any positive integer t , a t-broom is a graph obtained from K 1 , t + 1 by subdividing an edge once. In this paper, we show that, for graphs G without induced t -brooms, we have χ (G) = o (ω (G) t + 1) , where χ (G) and ω (G) are the chromatic number and clique number of G , respectively. When t = 2 , this answers a question of Schiermeyer and Randerath. Moreover, for t = 2 , we strengthen the bound on χ (G) to 7 ω (G) 2 , confirming a conjecture of Sivaraman. For t ≥ 3 and { t -broom, K t , t }-free graphs, we improve the bound to o (ω t). [ABSTRACT FROM AUTHOR]

Let 1 ⩽ a < b be two relatively prime integers. Sylvester found that a b − a − b is the largest integer which can not be represented by a x + b y (x , y ∈ Z ⩾ 0) about 160 years ago and this number shall be denoted by g a , b. Let N (a , b) = { n : n ⩽ g a , b , n = a x + b y , x , y ∈ Z ⩾ 0 } and π a , b be the number of primes in N (a , b). Recently, Ramírez Alfonsín and Skałba proved that π a , b ≫ ε g a , b (log ⁡ g a , b) 2 + ε for any fixed ε > 0. They further conjectured that the order of the magnitude of π a , b is 1 2 π (g a , b) , where π (x) is the number of all primes up to x. In this paper, we show that the conjecture is true for almost all pairs a , b with 1 ⩽ a < b and (a , b) = 1. The proofs rely heavily on the Bombieri–Vinogradov theorem and Brun–Titchmarsh theorem. [ABSTRACT FROM AUTHOR]

Let G be a K p -free graph. We say e is a K p -saturating edge of G if e ∉ E (G) and G + e contains a copy of K p. Denote by f p (n , m) the minimum number of K p -saturating edges that an n -vertex K p -free graph with m edges can have. Erdős and Tuza conjectured that f 4 (n , ⌊ n 2 / 4 ⌋ + 1) = (1 + o (1)) n 2 16. Balogh and Liu disproved this by showing f 4 (n , ⌊ n 2 / 4 ⌋ + 1) = (1 + o (1)) 2 n 2 33. They believed that a natural generalization of their construction for K p -free graph should also be optimal and made a conjecture that f p + 1 (n , ex (n , K p) + 1) = (2 (p − 2) 2 p (4 p 2 − 11 p + 8) + o (1)) n 2 for all integers p ≥ 3. The main result of this paper is to confirm the above conjecture of Balogh and Liu. [ABSTRACT FROM AUTHOR]

χ -bounded classes are studied here in the context of star colorings and, more generally, χ p -colorings. This fits to a general scheme of sparsity and leads to natural extensions of the notion of bounded expansion class. In this paper we solve two conjectures related to star coloring (i.e. χ 2) boundedness. One of the conjectures is disproved and in fact we determine which weakening holds true. χ p -boundedness leads to more stability and we give structural characterizations of (strong and weak) χ p -bounded classes. We also generalize a result of Wood relating the chromatic number of a graph to the star chromatic number of its 1-subdivision. As an application of our characterizations, among other things, we show that for every odd integer g > 3 even hole-free graphs G contain at most φ (g , ω (G)) | G | holes of length g. [ABSTRACT FROM AUTHOR]

For a number field F and an odd prime number p , let F ˜ be the compositum of all Z p -extensions of F and Λ ˜ the associated Iwasawa algebra. Let G S (F ˜) be the Galois group over F ˜ of the maximal extension which is unramified outside p -adic and infinite places. In this paper we study the Λ ˜ -module X S (− i) (F ˜) : = H 1 (G S (F ˜) , Z p (− i)) and its relationship with X (F ˜ (μ p)) (i − 1) Δ , the Δ : = Gal (F ˜ (μ p) / F ˜) -invariant of the Galois group over F ˜ (μ p) of the maximal abelian unramified pro- p -extension of F ˜ (μ p). More precisely, we show that under a decomposition condition, the pseudo-nullity of the Λ ˜ -module X (F ˜ (μ p)) (i − 1) Δ is implied by the existence of a Z p d -extension L with X S (− i) (L) : = H 1 (G S (L) , Z p (− i)) being without torsion over the Iwasawa algebra associated to L , and which contains a Z p -extension F ∞ satisfying H 2 (G S (F ∞) , Q p / Z p (i)) = 0. As a consequence we obtain a sufficient condition for the validity of Greenberg's generalized conjecture when the integer i ≡ 1 mod [ F (μ p) : F ]. This existence is fulfilled for (p , i) -regular fields. [ABSTRACT FROM AUTHOR]

Let F i denote the i th Fibonacci number, and define ∏ i = 1 n 1 + x F i + 1 = ∑ k c n (k) x k . The paper is concerned primarily with the coefficients c n (k). In particular, for any r ≥ 0 the generating function ∑ n ≥ 0 (∑ k c n (k) r) x n is rational. The coefficients c n (k) can be displayed in an array called the Fibonacci triangle poset F with some interesting further properties, including an encoding of a certain dense linear order on the nonnegative integers. Some generalizations are briefly considered, but there remain many open questions. [ABSTRACT FROM AUTHOR]

Let G = (Z / n Z) ⊕ (Z / n Z). Let s ≤ k (G) be the smallest integer ℓ such that every sequence of ℓ terms from G , with repetition allowed, has a nonempty zero-sum subsequence with length at most k. It is known that s ≤ 2 n − 1 − k (G) = 2 n − 1 + k for k ∈ [ 0 , n − 1 ]. The structure of extremal sequences that show this bound is tight was determined for k ∈ { 0 , 1 , n − 1 } , and for various special cases when k ∈ [ 2 , n − 2 ]. For the remaining values k ∈ [ 2 , n − 2 ] , the characterization of extremal sequences of length 2 n − 2 + k avoiding a nonempty zero-sum of length at most 2 n − 1 − k remained open in general. It is conjectured that they must all have the form e 1 [ n − 1 ] ⋅ e 2 [ n − 1 ] ⋅ (e 1 + e 2) [ k ] for some basis (e 1 , e 2) for G. Here x [ n ] denotes a sequence consisting of the term x repeated n times. In this paper, we establish this conjecture for all k ∈ [ 2 , n − 2 ] when n is prime, which in view of other recent work, implies the conjectured structure for all abelian groups of rank two. [ABSTRACT FROM AUTHOR]

In this paper, we prove a generalization of a conjecture of Erdös, about the chromatic number of certain Kneser-type hypergraphs. For integers n , k , r , s with n ≥ r k and 2 ≤ s ≤ r , the r -uniform general Kneser hypergraph KG s r (n , k) , has all k -subsets of { 1 , ... , n } as the vertex set and all multi-sets { A 1 , ... , A r } of k -subsets with s -wise empty intersections as the edge set. The case r = s = 2 , was considered by Kneser [7] in 1955, where he conjectured that its chromatic number is n − 2 (k − 1). This was finally proved by Lovász [9] in 1978. The case r > 2 and s = 2 , was considered by Erdös in 1973, and he conjectured that its chromatic number is ⌈ n − r (k − 1) r − 1 ⌉. This conjecture was proved by Alon, Frankl and Lovász [2] in 1986. The case where s > 2 , was considered by Sarkaria [11] in 1990, where he claimed to prove a lower bound for its chromatic number which generalized all previous results. Unfortunately, an error was found by Lange and Ziegler [14] in 2006 in the induction method of Sarkaria on the number of prime factors of r , and Sarkaria's proof only worked when s is less than the smallest prime factor of r or s = 2. Finally in 2019, Aslam, Chen, Coldren, Frick and Setiabrata [3] were able to extend this by using methods different from Sarkaria to the case when r = 2 α 0 p 1 α 1 ... p t α t and 2 ≤ s ≤ 2 α 0 (p 1 − 1) α 1 ... (p t − 1) α t . In this paper, by applying the Z p -Tucker lemma of Ziegler [13] and Meunier [10] , we finally prove the general Erdös conjecture and prove the claimed result of Sarkaria for any 2 ≤ s ≤ r. We also provide another proof of a special case of this result, using methods similar to those of Alon, Frankl, and Lovász [2] and compute the connectivity of certain simplicial complexes that might be of interest in their own right. [ABSTRACT FROM AUTHOR]

Let q = p e , where p is a prime and e is a positive integer. The family of graphs D (k , q) , defined for any positive integer k and prime power q , were introduced by Lazebnik and Ustimenko in 1995. To this day, the connected components of the graphs D (k , q) , provide the best known general lower bound for the size of a graph of given order and given girth. Furthermore, Ustimenko conjectured that the second largest eigenvalue of D (k , q) is always less than or equal to 2 q. If true, this would imply that for a fixed q and k growing, D (k , q) would define a family of expanders that are nearly Ramanujan. In this paper we prove the smallest open case of the conjecture, showing that for all odd prime powers q , the second largest eigenvalue of D (5 , q) is less than or equal to 2 q. [ABSTRACT FROM AUTHOR]

We prove a number of conjectures recently stated by P. Barry, related to the paperfolding sequence and the Rueppel sequence. Furthermore, we study the regularity of sequences involved in the paper, and prove that for all q ≥ 2 , the sequence consisting of the positive integers whose odd part is of the form 4 k + 1 is not q -regular. Finally we establish the 2-regularity of two sequences of Hankel determinants. [ABSTRACT FROM AUTHOR]

A folklore conjecture in number theory states that the only integers whose expansions in base 3 , 4 and 5 contain solely binary digits are 0 , 1 and 82000. In this paper, we present the first progress on this conjecture. Furthermore, we investigate the density of the integers containing only binary digits in their base 3 or 4 expansion, whereon an exciting transition in behaviour is observed. Our methods shed light on the reasons for this, and relate to several well-known questions, such as Graham's problem and a related conjecture of Pomerance. Finally, we generalise this setting and prove that the set of numbers in [ 0 , 1 ] who do not contain some digit in their b -expansion for all b ≥ 3 has zero Hausdorff dimension. [ABSTRACT FROM AUTHOR]

In 1983, Bouchet proposed a conjecture that every flow-admissible signed graph admits a nowhere-zero 6-flow. Bouchet himself proved that such signed graphs admit nowhere-zero 216-flows and Zýka further proved that such signed graphs admit nowhere-zero 30-flows. In this paper we show that every flow-admissible signed graph admits a nowhere-zero 11-flow. [ABSTRACT FROM AUTHOR]

The extremal number ex (n , F) of a graph F is the maximum number of edges in an n -vertex graph not containing F as a subgraph. A real number r ∈ [ 1 , 2 ] is realisable if there exists a graph F with ex (n , F) = Θ (n r). Several decades ago, Erdős and Simonovits conjectured that every rational number in [ 1 , 2 ] is realisable. Despite decades of effort, the only known realisable numbers are 0 , 1 , 7 5 , 2 , and the numbers of the form 1 + 1 m , 2 − 1 m , 2 − 2 m for integers m ≥ 1. In particular, it is not even known whether the set of all realisable numbers contains a single limit point other than the two numbers 1 and 2. In this paper, we make progress on the conjecture of Erdős and Simonovits. First, we show that 2 − a b is realisable for any integers a , b ≥ 1 with b > a and b ≡ ± 1 (mod a). This includes all previously known ones, and gives infinitely many limit points 2 − 1 m in the set of all realisable numbers as a consequence. Secondly, we propose a conjecture on subdivisions of bipartite graphs. Apart from being interesting on its own, we show that, somewhat surprisingly, this subdivision conjecture in fact implies that every rational number between 1 and 2 is realisable. [ABSTRACT FROM AUTHOR]

In this paper, using quaternion arithmetic in the ring of Lipschitz integers, we present a proof of Zhi-Wei Sun's "1-3-5 conjecture" for all integers, and reduce the general case to its verification up to 1.052 × 10 11. The computational verification was performed by the authors and a colleague, concluding the proof of Sun's 1-3-5 conjecture. We also establish some variations of this conjecture. [ABSTRACT FROM AUTHOR]

Let n be a fixed integer with n ≥ 2. For i , j ∈ Z n , define | | i , j | | n to be the distance between i and j when the elements of Z n are written in a cycle. So | | i , j | | n = min ⁡ { (i − j) mod n , (j − i) mod n }. For positive integers s and k , the permutation π : Z n → Z n is (s , k) -clash-free if | | π (i) , π (j) | | n ≥ k whenever | | i , j | | n < s with i ≠ j. So an (s , k) -clash-free permutation π can be thought of as moving every close pair of elements of Z n to a pair at large distance. More geometrically, the existence of an (s , k) -clash-free permutation is equivalent to the existence of a set of n non-overlapping s × k rectangles on an n × n torus, whose centres have distinct integer x -coordinates and distinct integer y -coordinates. For positive integers n and k with k < n , let σ (n , k) be the largest value of s such that an (s , k) -clash-free permutation on Z n exists. In a recent paper, Mammoliti and Simpson conjectured that ⌊ (n − 1) / k ⌋ − 1 ≤ σ (n , k) ≤ ⌊ (n − 1) / k ⌋ for all integers n and k with k < n. The paper establishes this conjecture, by explicitly constructing an (s , k) -clash-free permutation on Z n with s = ⌊ (n − 1) / k ⌋ − 1. Indeed, this construction is used to establish a more general conjecture of Mammoliti and Simpson, where for some fixed integer r we require every point on the torus to be contained in the interior of at most r rectangles. [ABSTRACT FROM AUTHOR]

Let f r (n , v , e) denote the maximum number of edges in an r -uniform hypergraph on n vertices, in which the union of any e distinct edges contains at least v + 1 vertices. The study of f r (n , v , e) was initiated by Brown, Erdős and Sós more than forty years ago. In the literature, the following conjecture is well known. Conjecture: n k − o (1) < f r (n , e r − (e − 1) k + 1 , e) = o (n k) holds for all fixed integers r > k ≥ 2 and e ≥ 3 as n → ∞. For r = 3 , e = 3 , k = 2 , the bound n 2 − o (1) < f 3 (n , 6 , 3) = o (n 2) was proved by the celebrated (6,3)-theorem of Ruzsa and Szemerédi. In this paper, we add more evidence for the validity of the conjecture. On one hand, using the hypergraph removal lemma we show that the upper bound part of the conjecture is true for all fixed integers r ≥ k + 1 ≥ e ≥ 3. On the other hand, using tools from additive number theory we present several constructions showing that the lower bound part of the conjecture is true for r ≥ 3 , k = 2 and e = 4 , 5 , 7 , 8. Prior to our results, all known constructions that match the conjectured lower bound satisfy either r = 3 or e = 3. Our constructions are the first ones in the literature that break this barrier. [ABSTRACT FROM AUTHOR]

In this paper we study the following eigenvalue problem { − Δ v = λ C (α) (p α − ε) | x | α u ε p α − ε − 1 v in Ω , u = 0 on ∂ Ω , where Ω ⊂ R N is a smooth bounded domain containing the origin, C (α) = (N + α) (N − 2) , N ≥ 3 , p α = N + 2 + 2 α N − 2 , α > 0 , ε > 0 is a small parameter and u ε is a single peaked solution of Hénon equation { − Δ u = C (α) | x | α u p α − ε in Ω , u > 0 in Ω , u = 0 on ∂ Ω , which established by Gladiali and Grossi (2012) [21]. By using various local Pohozaev identities and blow-up analysis, we prove some asymptotic behavior of the eigenvalues λ ε , i and corresponding eigenfunctions v ε , i , i = 2 , ⋯ , ∑ 1 ≤ k < 2 + α 2 (N + 2 k − 2) (N + k − 3) ! (N − 2) ! k ! + 2 when α is not an even integer. As a consequence, if 0 < α < 2 , we have that the Morse index of the single peaked solutions is N + 1 , which gives an affirmative answer to a conjecture raised by Gladiali and Grossi. [ABSTRACT FROM AUTHOR]

A singular k -tuple of a tensor T of format (n 1 , ... , n k) is essentially a complex critical point of the distance function from T constrained to the cone of tensors of format (n 1 , ... , n k) of rank at most one. A generic tensor has finitely many complex singular k -tuples, and their number depends only on the tensor format. Furthermore, if we fix the first k − 1 dimensions n i , then the number of singular k -tuples of a generic tensor becomes a monotone non-decreasing function in one integer variable n k , that stabilizes when (n 1 , ... , n k) reaches a boundary format. In this paper, we study the linear span of singular k -tuples of a generic tensor. Its dimension also depends only on the tensor format. In particular, we concentrate on special order three tensors and order- k tensors of format (2 , ... , 2 , n). As a consequence, if again we fix the first k − 1 dimensions n i and let n k increase, we show that in these special formats, the dimension of the linear span stabilizes as well, but at some concise non-sub-boundary format. We conjecture that this phenomenon holds for an arbitrary format with k > 3. Finally, we provide equations for the linear span of singular triples of a generic order three tensor T of some special non-sub-boundary format. From these equations, we conclude that T belongs to the linear span of its singular triples, and we conjecture that this is the case for every tensor format. [ABSTRACT FROM AUTHOR]

In 1927, Artin hypothesized that for any given non-zero integer a other than 1, −1, or a perfect square, there exists infinitely many primes p for which a is a primitive root modulo p. In 1967, Hooley proved it under the assumption of the generalized Riemann hypothesis. Since then, there are many analogues and generalization of this conjecture. In this paper, we work on its generalization to composite moduli in the function fields setting. Let A = F q t be the ring of polynomials over the finite field F q and 0 ≠ a ∈ A. Let C be the A -Carlitz module. Let a be a fixed element in A. For n ∈ A , C (A / n A) is a finite A -module. The set of all annihilators of C (A / n A) is an ideal and generated by a monic polynomial, denoted by λ (n). Similarly, The set of all annihilators of the submodule of C (A / n A) generated by a is an ideal and let l a (n) be its monic generator. We say that a is a primitive root of n , if λ (n) = l a (n). Define N a (x) : = | { n ∈ A | deg ⁡ n = x , n is monic , a is a primitive root of n } | We prove that for a given non-constant a ∈ A , a ∉ E , an exceptional set, there exists an unbounded set V of integers such that lim inf x ∈ V N a (x) / q x = 0 This result is analogous to Li's theorem for Artin's conjecture on composite moduli. It is the first time that this kind of results holds in the setting of the function fields. [ABSTRACT FROM AUTHOR]

Twenty years ago Bondy and Vince conjectured that for any nonnegative integer k , except finitely many counterexamples, every graph with k vertices of degree less than three contains two cycles whose lengths differ by one or two. The case k ≤ 2 was proved by Bondy and Vince, which resolved an earlier conjecture of Erdős et al. In this paper we confirm this conjecture for all k. [ABSTRACT FROM AUTHOR]

Denote by P the set of all primes and by P + (n) the largest prime factor of integer n ⩾ 1 with the convention P + (1) = 1. For each η > 1 , let c = c (η) > 1 be some constant depending on η and P a , c , η : = { p ∈ P : p = P + (q − a) for some prime q with p η < q ⩽ c (η) p η }. In this paper, under the Elliott-Halberstam conjecture we prove, for y → ∞ , π a , c , η (x) : = | (1 , x ] ∩ P a , c , η | ∼ π (x) or π a , c , η (x) ≫ a , η π (x) according to values of η. These are complement for some results of Banks-Shparlinski [1] , of Wu [12] and of Chen-Wu [2]. [ABSTRACT FROM AUTHOR]

Let G be a graph with possible multiple edges but no loops. The density of G , denoted by ρ (G) , is defined as max H ⊂ G , | V (H) | ≥ 2 ⁡ ⌈ | E (H) | ⌊ | V (H) | / 2 ⌋ ⌉. Goldberg (1973) and Seymour (1974) independently conjectured that if the chromatic index χ ′ (G) satisfies χ ′ (G) ≥ Δ (G) + 2 then χ ′ (G) = ρ (G) , which is commonly regarded as Goldberg's conjecture. An equivalent conjecture, usually credited to Jakobsen, states that for any odd integer m ≥ 3 , if χ ′ (G) ≥ m Δ (G) m − 1 + m − 3 m − 1 then χ ′ (G) = ρ (G). The Tashkinov tree technique, a common generalization of Vizing fans and Kierstead paths for multigraphs, has emerged as the main tool to attack these two conjectures. On the other hand, Asplund and McDonald recently showed that there is a limitation to this method. In this paper, we will go beyond Tashkinov trees and provide a much larger extended structure, using which we see hope to tackle the conjecture. Applying this new technique, we show that the Goldberg's conjecture holds for graphs with Δ (G) ≤ 39 or | V (G) | ≤ 39 and the Jakobsen Conjecture holds for m ≤ 39 , where the previously known best bound is 23. We also improve a number of other related results. [ABSTRACT FROM AUTHOR]

Let V ( D ) = X ∪ Y be a bipartition of a directed graph D . We use e ( X , Y ) to denote the number of arcs in D from X to Y . Motivated by a conjecture posed by Lee, Loh and Sudakov (2016) [16] , we study bipartitions of oriented graphs. Let D be an oriented graph with m arcs. In this paper, it is proved that if the minimum degree of D is δ , then D admits a bipartition V ( D ) = V 1 ∪ V 2 such that min ⁡ { e ( V 1 , V 2 ) , e ( V 2 , V 1 ) } ≥ ( δ − 1 4 δ + o ( 1 ) ) m . Moreover, if the minimum semidegree d = min ⁡ { δ + ( D ) , δ − ( D ) } of D is at least 21, then D admits a bipartition V ( D ) = V 1 ∪ V 2 such that min ⁡ { e ( V 1 , V 2 ) , e ( V 2 , V 1 ) } ≥ ( d 2 ( 2 d + 1 ) + o ( 1 ) ) m . Both bounds are asymptotically best possible. [ABSTRACT FROM AUTHOR]

For graphs H and F with chromatic number χ (F) = k , we call H strictly F -Turán-good (or (H , F) strictly Turán-good) if the Turán graph T k − 1 (n) is the unique F -free graph on n vertices containing the largest number of copies of H when n is large enough. Let F be a graph with chromatic number χ (F) ≥ 3 and a color-critical edge and let P ℓ be a path with ℓ vertices. Gerbner and Palmer (2020) showed that (P 3 , F) is strictly Turán good if χ (F) ≥ 4 and they conjectured that (a) this result is true when χ (F) = 3 , and, moreover, (b) (P ℓ , K k) is Turán-good for every pair of integers ℓ and k. In the present paper, we show that (H , F) is strictly Turán-good when H is a bipartite graph with matching number ν (H) = ⌊ | V (H) | 2 ⌋ and χ (F) = 3 , as a corollary, this result confirms the conjecture (a); we also prove that (P ℓ , F) is strictly Turán-good for 2 ≤ ℓ ≤ 6 and χ (F) ≥ 4 , this also confirms the conjecture (b) for 2 ≤ ℓ ≤ 6 and k ≥ 4. [ABSTRACT FROM AUTHOR]

Text Let N be the set of all nonnegative integers and k ≥ 2 be a fixed integer. For a set A ⊆ N , let r k ( A , n ) denote the number of solutions of a 1 + ⋯ + a k = n with a 1 , … , a k ∈ A . In this paper, we prove that for given positive integer u , there is a set A ⊆ N such that r k ( A , n ) ≥ 1 for all n ≥ 0 and the set of n with r k ( A , n ) = k ! u has density one. This generalizes recent results of Chen and Yang. Video For a video summary of this paper, please visit http://youtu.be/2fbKtDAOqQ0 . [ABSTRACT FROM AUTHOR]

It was conjectured by Jaeger that every 4 p-edge-connected graph admits a modulo ( 2 p + 1 ) -orientation (and, therefore, admits a nowhere-zero circular ( 2 + 1 p ) -flow). This conjecture was partially proved by Lovász et al. (2013) [7] for 6 p -edge-connected graphs. In this paper, infinite families of counterexamples to Jaeger's conjecture are presented. For p ≥ 3 , there are 4 p -edge-connected graphs not admitting modulo ( 2 p + 1 ) -orientation; for p ≥ 5 , there are ( 4 p + 1 ) -edge-connected graphs not admitting modulo ( 2 p + 1 ) -orientation. [ABSTRACT FROM AUTHOR]

Let N ( a , b , c , d ; n ) and t ( a , b , c , d ; n ) denote the number of representations of n as a x 2 + b y 2 + c z 2 + d w 2 and the number of representations of n as a x ( x + 1 ) 2 + b y ( y + 1 ) 2 + c z ( z + 1 ) 2 + d w ( w + 1 ) 2 , respectively, where a , b , c , d are positive integers, n is a nonnegative integer and x , y , z , w are integers. Sun established many relations between N ( a , b , c , d ; n ) and t ( a , b , c , d ; n ) and posed 23 conjectures. Yao proved five of them by using ( p , k ) -parametrization of theta functions. In this paper, we confirm four conjectures of Sun by employing theta function identities. [ABSTRACT FROM AUTHOR]

Hadwiger's conjecture from 1943 states that for every integer t ≥ 1 , every graph either can be t -colored or has a subgraph that can be contracted to the complete graph on t + 1 vertices. As pointed out by Paul Seymour in his recent survey on Hadwiger's conjecture, proving that graphs with no K 7 minor are 6-colorable is the first case of Hadwiger's conjecture that is still open. It is not known yet whether graphs with no K 7 minor are 7-colorable. Using a Kempe-chain argument along with the fact that an induced path on three vertices is dominating in a graph with independence number two, we first give a very short and computer-free proof of a recent result of Albar and Gonçalves and generalize it to the next step by showing that every graph with no K t minor is ( 2 t − 6 ) -colorable, where t ∈ { 7 , 8 , 9 } . We then prove that graphs with no K 8 − minor are 9-colorable, and graphs with no K 8 = minor are 8-colorable. Finally we prove that if Mader's bound for the extremal function for K t minors is true, then every graph with no K t minor is ( 2 t − 6 ) -colorable for all t ≥ 6 . This implies our first result. We believe that the Kempe-chain method we have developed in this paper is of independent interest. [ABSTRACT FROM AUTHOR]

Aharoni and Howard conjectured that, for positive integers n , k , t with n ≥ k t , if F 1 , ... , F t ⊆ ( [ n ] k ) such that | F i | > max ⁡ { ( n k ) − ( n − t + 1 k ) , ( k t − 1 k ) } for i ∈ [ t ] then there exist e i ∈ F i for i ∈ [ t ] such that e 1 , ... , e t are pairwise disjoint. Huang, Loh, and Sudakov proved this conjecture for t < n / (3 k 2). In this paper, we show that this conjecture holds for t < n / (2 k) and t sufficiently large. [ABSTRACT FROM AUTHOR]

The Markov numbers are the positive integers that appear in the solutions of the equation x 2 + y 2 + z 2 = 3 x y z. These numbers are a classical subject in number theory and have important ramifications in hyperbolic geometry, algebraic geometry and combinatorics. It is known that the Markov numbers can be labeled by the lattice points (q , p) in the first quadrant and below the diagonal whose coordinates are coprime. In this paper, we consider the following question. Given two lattice points, can we say which of the associated Markov numbers is larger? A complete answer to this question would solve the uniqueness conjecture formulated by Frobenius in 1913. We give a partial answer in terms of the slope of the line segment that connects the two lattice points. We prove that the Markov number with the greater x -coordinate is larger than the other if the slope is at least − 8 7 and that it is smaller than the other if the slope is at most − 5 4. As a special case, namely when the slope is equal to 0 or 1, we obtain a proof of two conjectures from Aigner's book "Markov's theorem and 100 years of the uniqueness conjecture". [ABSTRACT FROM AUTHOR]

It was conjectured in the monograph [9] by Godsil and Meagher and in the article [10] by Lindzey that the perfect matching derangement graph M 2 n possesses the alternating sign property, that is, for any integer partition λ = (λ 1 , ... , λ r) ⊢ n , the sign of the eigenvalue η λ of M 2 n is given by sign (η λ) = (− 1) n − λ 1 . In this paper, we prove that the conjecture is true. Our approach yields a recurrence formula for the eigenvalues of the perfect matching derangement graph as well as a new recurrence formula for the eigenvalues of the permutation derangement graph. [ABSTRACT FROM AUTHOR]

Abstract: Let be a prime, and let m be an integer with . In the paper we solve some conjectures of Z.W. Sun concerning , and . In particular, we show that for . Let be the Legendre polynomials. In the paper we also show that , where t is a rational p-adic integer, is the greatest integer not exceeding x and is the Legendre symbol. As consequences we determine in the cases and confirm many conjectures of Z.W. Sun. [Copyright &y& Elsevier]

We say that a permutation π = π 1 π 2 ⋯ π n ∈ S n has a peak at index i if π i − 1 < π i > π i + 1 . Let P ( π ) denote the set of indices where π has a peak. Given a set S of positive integers, we define P ( S ; n ) = { π ∈ S n : P ( π ) = S } . In 2013 Billey, Burdzy, and Sagan showed that for subsets of positive integers S and sufficiently large n , | P ( S ; n ) | = p S ( n ) 2 n − | S | − 1 where p S ( x ) is a polynomial depending on S . They proved this by establishing a recursive formula for p S ( x ) involving an alternating sum, and they conjectured that the coefficients of p S ( x ) expanded in a binomial coefficient basis centered at max ⁡ ( S ) are all nonnegative. In this paper we introduce a new recursive formula for | P ( S ; n ) | without alternating sums and we use this recursion to prove that their conjecture is true. [ABSTRACT FROM AUTHOR]

In this paper we deal with Diophantine equations involving products of consecutive integers, inspired by a question of Erdős and Graham. [ABSTRACT FROM AUTHOR]

A graph Γ is said to be distance-balanced if for any edge u v of Γ , the number of vertices closer to u than to v is equal to the number of vertices closer to v than to u , and it is called nicely distance-balanced if in addition this number is independent of the chosen edge u v. A graph Γ is said to be strongly distance-balanced if for any edge u v of Γ and any integer k , the number of vertices at distance k from u and at distance k + 1 from v is equal to the number of vertices at distance k + 1 from u and at distance k from v. In this paper we solve an open problem posed by Kutnar and Miklavič (2014) by constructing several infinite families of nonbipartite nicely distance-balanced graphs which are not strongly distance-balanced. We disprove a conjecture regarding characterization of strongly distance-balanced graphs posed by Balakrishnan et al. (2009) by providing infinitely many counterexamples, and answer a question posed by Kutnar et al. in (2006) regarding the existence of semisymmetric distance-balanced graphs which are not strongly distance-balanced by providing an infinite family of such examples. We also show that for a graph Γ with n vertices and m edges it can be checked in O (m n) time if Γ is strongly-distance balanced and if Γ is nicely distance-balanced. [ABSTRACT FROM AUTHOR]

Let P + (n) denote the largest prime factor of an integer n. In this paper, we study the asymptotics of sums ∑ p ⩽ x P + (p + h) ⩽ y 1 and ∑ n ⩽ x , P + (n) ⩽ y 1 P + (n + h) ⩽ y 2 1 for x ε ⩽ y , y 1 , y 2 ⩽ x. Our results imply that two conjectures concerning the asymptotics for the above two sums, which were considered by Erdős and Pomerance, hold on average. As an application of the first sum above, we can deduce that there are infinitely many primes p ⩽ x such that P + (p + h) > p 1 − ε for all but o ((log ⁡ x) c 0 ) values of shifts h , where c 0 is a small computable constant. As applications of the second sum above, we consider the averaged version of Erdős–Turán's conjecture and the equation a + b = c. In particular, we show that there exists at least a positive density (>17.505%) of triples of integers { a , b , c } such that a + b = c and c ≪ P + (a) P + (c). This upper bound is stronger than c ≪ (∏ p | a b c p) 1 + ε in the abc -conjecture. [ABSTRACT FROM AUTHOR]

It is conjectured by Berge and Fulkerson that every bridgeless cubic graph has six perfect matchings such that each edge is contained in exactly two of them. This conjecture has been verified for many families of snarks with small (≤ 5) cyclic edge-connectivity. An infinite family, denoted by S K , of cyclically 6-edge-connected superposition snarks was constructed in [European J. Combin. 2002] by Kochol. In this paper, the Berge–Fulkerson conjecture is verified for the family S K , and, furthermore, some larger families containing S K. This is the first paper about the Berge–Fulkerson conjecture for superposition snarks and cyclically 6-edge-connected snarks. Tutte's integer flow and Catlin's contractible configuration are applied here as the key methods. [ABSTRACT FROM AUTHOR]

Let G be an additive finite abelian group of exponent exp ⁡ (G). For every positive integer k , let s k exp ⁡ (G) (G) denote the smallest integer t such that every sequence over G of length t has a zero-sum subsequence of length k exp ⁡ (G). Let η k exp ⁡ (G) (G) denote the smallest integer t such that every sequence over G of length t has a zero-sum subsequence of length between 1 and k exp ⁡ (G). It is conjectured by Gao et al. that s k exp ⁡ (G) (G) = η k exp ⁡ (G) (G) + k exp ⁡ (G) − 1 for all pairs of (G , k). This conjecture is a common generalization of several previous conjectures and has been confirmed for some special pairs of (G , k). In this paper we shall prove this conjecture for more pairs of (G , k). We also study the inverse problem associated with s k exp ⁡ (G) (G) , i.e., we determine the structure of sequences S of length s k exp ⁡ (G) (G) − 1 that have no zero-sum subsequence of length k exp ⁡ (G). [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we show that for given positive integer C, there are only finitely many distance-regular graphs with valency k at least three, diameter D at least six and . This extends a conjecture of Bannai and Ito. [Copyright &y& Elsevier]

Abstract: For any set of nonnegative integers, let be the number of solutions to the equation . The set is called a basis of if for all . The well known Erdős–Turán conjecture says that if is a basis of , then cannot be bounded. In 1990, Ruzsa proved that there exists a basis of such that . In this paper, we give a new proof of Ruzsa’s Theorem. [Copyright &y& Elsevier]

Abstract: For any positive integer n we state and prove formulas for the number of solutions, in integers, of , , and . Some conjectures are listed at the end of the paper. [Copyright &y& Elsevier]

Abstract: In 1956 L. Jeśmanowicz conjectured, for any primitive Pythagorean triple satisfying , that the equation has the unique solution in positive integers x, y and z. This is a famous unsolved problem on Pythagorean numbers. In this paper we broadly extend many of classical well-known results on the conjecture. As a corollary we can verify that the conjecture is true if . [Copyright &y& Elsevier]

Abstract: Text: Let be the least number of ones that can be used to represent n using ones and any number of + and × signs (and parentheses). It is always true that for a prime p, Itʼs in the famous book Unsolved Problems in Number Theory as problem F26 written by R.K. Guy. The aim of our work is to give a counterexample to the conjecture and some properties on . Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=R8IQI_dwaJE. [Copyright &y& Elsevier]

Abstract: In 1947, Lehmer conjectured that the Ramanujan τ-function is non-vanishing for all positive integers m, where are the Fourier coefficients of the cusp form Δ of weight 12. It is known that Lehmerʼs conjecture can be reformulated in terms of spherical t-design, by the result of Venkov. In this paper, we show that is equivalent to the fact that the homogeneous space of the moonshine vertex operator algebra is a conformal 12-design. Therefore, Lehmerʼs conjecture is now reformulated in terms of conformal t-designs. [Copyright &y& Elsevier]

Abstract: In 1992, Strauss, Shallit and Zagier proved that for any positive integer a, and furthermore Recently a q-analogue of the first congruence was conjectured by Guo and Zeng. In this paper we prove the conjecture of Guo and Zeng, and also give a q-analogue of the second congruence. [Copyright &y& Elsevier]

Abstract: It is conjectured that all sufficiently large integers satisfying some necessary congruence conditions are the sum of four cubes of primes. In this paper, it is proved that the conjecture is true for at least 2.911% of the positive integers satisfying the necessary conditions. This improves the result 1.5% due to Ren (2003) . [Copyright &y& Elsevier]